Quantum Entanglement of Local Operators

J. G. Cramer

Zhaofeng Su, Nina Sukhodoeva

Quantum entanglement is an indispensable resource for many significant quantum information processing tasks. Thus, distilling more entanglement from less entangled resource is a task of practical significance and has been investigated for decades. The literature [Verstraete \textit{et al}., \href{https://link.aps.org/doi/10.1103/PhysRevA.64.010101}{Phys. Rev. A 64, 010101(2001)}] considered a scenario to increase the entanglement by local filtering operation and qualitatively derived the variance relation of entanglement. We investigate the scenario with general two-qubit resources to find the optimal strategy of filtering operations. We obtain the upper bound for the ratio of entanglement increase and find the corresponding optimal local filtering operation to achieve the maximal ratio. Our analysis shows that the upper bound ratio grows with the length of local Bloch vector while the success probability decrease with it. We further extend the research to investigate the optimal measurement strategy by considering general measurement. Our result shows that local measurement can not increase the expectation of quantum entanglement, which gives more analytical evidence to the well known fact that local operation can not create quantum entanglement.

Ming-Yong Ye, Yong-Sheng Zhang, Guang‐Can Guo

Nina Sukhodoeva, Zhaofeng Su

Entanglement is a remarkable phenomenon observed in quantum systems, which has many potential applications in the field of computing, communications and cryptography. In quantum information theory, entanglement is considered as an indispensable resource that can be quantified and purified. In typical quantum information task, a quantum state is distributed between two spatially separated observers that can only perform operations on their part of quantum system. In this scenario, the possibility of increasing amount of entanglement is of practical importance. In this paper we investigate the increase of quantum entanglement that can be achieved by local operations for 2-qubit states. Firstly, we employ the formula for the concurrence change under local operations to analyze the possibility of entanglement increase for a pair of local operations depending on the parameters of operators and the state. Secondly, we evaluate the expectation value of the concurrence under a set of local operations performed by one and two observers. Finally, we consider two classes of density matrices under stochastic local operations and classical communication.

Gus Gutoski

Multi-party local quantum operations with shared quantum entanglement or shared classical randomness are studied. The following facts are established: * There is a ball of local operations with shared randomness lying within the space spanned by the no-signaling operations and centred at the completely noisy channel. * The existence of the ball of local operations with shared randomness is employed to prove that the weak membership problem for local operations with shared entanglement is strongly NP-hard. * Local operations with shared entanglement are characterized in terms of linear functionals that are "completely'' positive on a certain cone K of separable Hermitian operators, under a natural notion of complete positivity appropriate to that cone. Local operations with shared randomness (but not entanglement) are also characterized in terms of linear functionals that are merely positive on that same cone K. * Existing characterizations of no-signaling operations are generalized to the multi-party setting and recast in terms of the Choi-Jamio\l kowski representation for quantum super-operators. It is noted that the standard nonlocal box is an example of a no-signaling operation that is separable, yet cannot be implemented by local operations with shared entanglement.