Entanglement transformation between ensembles of three-qubit GHZ-type states by LOCC

Dechao Li

Jiayang Xiao

This thesis studies the manipulation of entanglement in three-qubit quantum systems. I consider the operational setting in which the qubits are distributed to three spatially separated parties. The parties act locally on their quantum systems and share classical communication with one another, a scenario commonly called local operations and classical communication (LOCC). In the LOCC setting, there are two different classes of entanglement in multipartite systems, called the GHZ and W classes, which are inequivalent in the sense that states from one class cannot be transformed into the other without the consumption of additional entanglement. In this thesis, I first show that the LOCC conversion of certain GHZ and W-class states becomes possible by using only one additional ebit (“entangled bit”) of shared entanglement. In many cases, this can be proven as the minimal amount of needed entanglement. I then consider the problem of one-way communication transformations from general three-qubit states into two-qubit maximally entangled states, known as EPR states. An optimal protocol in terms of success probability is provided for W-class states. The success probability of this protocol coincides with the optimal success probability if two of the parties are allowed to act jointly within the same laboratory. In other words, forcing the locality constraint on all three parties does not weaken their capabilities for obtaining bipartite entanglement when starting from a W-class state. I also present that this property holds for certain types of GHZ-class states as well.

Toshiyuki Tashima, T. Wakatsuki, S. K. Ozdemi +3 more

In quantum information science, it is often the question what distantly located parties sharing EPR pairs can achieve if they are restricted to only local operations and classical communication (LOCC). State preparation and transformation is one of such tasks, which have been attracted considerable interest since the early years of the field. It is well-known that starting with EPR pairs, any two-qubit pure state can be prepared via only LOCC. Thus EPR pairs form a universal resource. However, the situation is very different when the multipartite entanglement is considered, because in multipartite entanglement, there are distinct classes of entanglement, which cannot be converted into each other with unit fidelity even under stochastic local operation and classical communication (SLOCC), e.g., two distinct classes of states, GHZ-type and W-type states for the tripartite cases [1]. There exists no tripartite state that can be used as a universal resource to generate arbitrary tripartite pure states using SLOCC. For this purpose, one must look for a resource in larger systems such as many shared EPR pairs to extract the desired multipartite state. For example, there is a direct way of converting two EPR pairs into a three-qubit GHZ-type state [2], which then can be used to prepare approximate W state [3]. In this conversion from GHZ to W state, there is a trade-off between the fidelity and success probability. Unit fidelity cannot be achieved with non-zero probability, which confirms that GHZ- and W-type states belong to distinct classes.

Jens Siewert, Robert Lohmayer, Armin Uhlmann +1 more

Gökhan Torun, Ali Kerem Yildiz

The states of three-qubit systems split into two inequivalent types of genuine tripartite entanglement, namely, the Greenberger-Horne-Zeilinger (GHZ) type and the $W$ type. A state belonging to one of these classes can be stochastically transformed only into a state within the same class by local operations and classical communications. We provide local quantum operations, consisting of the most general two-outcome measurement operators, for the deterministic transformations of three-qubit pure states in which the initial and the target states are in the same class. We explore these transformations, originally having standard GHZ and standard $W$ states, under the local measurement operations carried out by a single party and $p$ ($p=2,3$) parties (successively). We find a notable result that the standard GHZ state cannot be deterministically transformed to a GHZ-type state in which all its bipartite entanglements are nonzero, i.e., a transformation can be achieved with unit probability when the target state has at least one vanishing bipartite concurrence.

Yong Wook Cheong, Seung-Woo Lee, Jinhyoung Lee