Entanglement measures based on the complete information of reduced states

Quantum entanglement has been verified experimentally and applied in quantum computing, quantum sensing, and quantum networks. It is of great significance to find measures to characterize the quantum entanglement faithfully. In this work, by exploiting the Schmidt decomposition of bipartite states, we first establish a one-to-one correspondence between the characteristic polynomial of the reduced state of a bipartite pure state and the trace of the reduced state. We introduce a family of entanglement measures based on the eigenvalues of the reduced density matrices. Specific measures called informationally complete entanglement measures (ICEMs) are presented. It is demonstrated that such ICEMs can characterize better the entanglement than the existing well-known entanglement measures. The ICEMs also give rise to criteria of state transformations under local operation and classical communication. Moreover, it is shown that the ICEMs can be efficiently estimated on a quantum computer. The full separability, entanglement, and genuine multipartite entanglement can be detected faithfully on quantum devices.

Figure 1

Advantage of ICEMs for distinguishing states. The blue dot-dashed line is the ICEM of $|{\varphi }_1{\rangle }_{AB}$ and the red solid line represents the ICEM of $|\varphi \rangle$, as a function of the states given by the points on the ellipse.

Figure 2

Circuit for calculating $E^{\mathcal{C}}{\left(\right|\psi \rangle }_{AB})$. Given $r+1$ copies of the quantum state ${|\psi \rangle }_{AB}$ and $r$ ancilla qubits, the $r$ ancilla qubits are applied one by one by a swap test by employing the $k\mathrm{th}$ ($k=1,2,...,r$) ancilla to perform a controlled swap test on the $k\mathrm{th}$ and $(k+1)\mathrm{th}$ copies of ${|\psi \rangle }_{AB}$.