Minimal control power of controlled dense coding and genuine tripartite entanglement

Abstract

We investigate minimal control power (MCP) for controlled dense coding defined by the channel capacity. We obtain MCPs for extended three-qubit Greenberger-Horne-Zeilinger (GHZ) states and generalized three-qubit W states. Among those GHZ states, the standard GHZ state is found to maximize the MCP and so does the standard W state among the W-type states. We find the lower and upper bounds of the MCP and show for pure states that the lower bound, zero, is achieved if and only if the three-qubit state is biseparable or fully separable. The upper bound is achieved only for the standard GHZ state. Since the MCP is nonzero only when three-qubit entanglement exists, this quantity may be a good candidate to measure the degree of genuine tripartite entanglement.Superdense coding 1 is one of the simplest examples showing the power of quantum entanglement. It allows one to transmit two bits of classical information by sending only one qubit if a maximally entangled state is initially shared by the sender and receiver. It has been studied theoretically 2-9 as well as experimentally [10][11][12] .Quantum teleportation 13 is another intriguing example utilizing the power of quantum entanglement. It is a protocol to transmit an unknown quantum state using classical communication and an initially shared maximally entangled state. In fact, the equivalence of quantum teleportation and superdense coding with maximally entangled states has been proven 14 .Controlled dense coding 15,16 , and controlled quantum teleportation 17 have been proposed as extensions of superdense coding and quantum teleportation. The standard dense coding and quantum teleportation involve only two parties, a sender and a receiver, and assume that they share a maximally entangled state in advance. In controlled dense coding and teleportation, a third party participates in the protocol as a controller, and a tripartite entangled state is shared among the three parties. The controller in each protocol can control the channel capacity and the teleportation fidelity of the other two parties, respectively. Recently, the concepts of control power (CP) and minimal control power (MCP) of the controlled teleportation have been suggested to quantify how much teleportation fidelity can be controlled by the controller [18][19][20][21] . CP is defined as the difference between teleportation fidelities with and without the controller's assistance. MCP is defined as the minimal value of CP among all possible permutations of the three qubits. Similarly, CP has been defined in controlled remote state preparation schemes as well 22 .Motivated by the similarity between quantum teleportation and superdense coding, we here define CP and MCP of controlled dense coding of three-qubit states to quantify how much the dense coding channel capacity can be controlled by the controller. We also calculate CP and MCP for two representative tripartite entangled states: the extended Greenberger-Horne-Zeilinger (GHZ) state and the generalized W state. We show that the standard GHZ state an...

Generalization of port-based teleportation and controlled teleportation capability

“…First, we can imagine another variants of the PBT protocols or their communication capabilities in quantum communication. For example, a controlled densecoding [31] and a remote state-preparation scheme [32]. Since the performance of the PBT has been studied in a probabilistic scenario as well as in a deterministic one, we can also investi-gate how the controlled version behaves in a probabilistic scenario.…”

Optimal quantum channel for perfect controlled super-dense coding protocol

“…To assess the role of the controller, we exploit the concept of the so-called controller's power (or just control power for short) CP which was defined previously in [41] as the difference between the number of cbits that can be transmitted with and without the controller's cooperation, i.e., CP I S 2 . 9…”

“…Such an aspect of control power has still been little investigated. For example, the role of the controller in controlled super-dense coding was studied recently in [41] with regard to the minimal control power.…”

“…The controlled super-dense coding was experimentally demonstrated by Jing et al [21] and Zhang et al [22]. Later, researchers have generalized the controlled super-dense coding protocols to multi-partite quantum channels to extend various coding tasks within a quantum network [23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41]. The quantum channel of the controlled super-dense coding protocol may be an entangled state of different kinds such as Greenberger-Horne-Zeilinger (GHZ) state [23], GHZ-class state [24], extended GHZ state [26], four-partite non-maximally entangled state [27], five-qubit cluster state [28], generalized GHZ-type state [29], six-qubit cluster state [30], a genuine five-atom entangled state in cavity QED [34] and so on.…”