Minimal control power of controlled dense coding and genuine tripartite entanglement

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.

the sender, Alice, and the receiver, Bob, initially share a Bell state, and Alice encodes classical information by performing a local operation on her qubit and then sends it to Bob. After receiving her qubit, Bob performs a twoqubit measurement so that he can recover the classical information that Alice wants to transmit. Thus, Alice is able to transmit two classical bits by sending one qubit of a Bell state that is initially shared between them. Indeed, partially entangled states can be used in the dense coding scheme instead of maximally entangled states, and the number of bits that Alice can transmit to Bob using general two-qubit state ρ 12 can be quantified by the dense coding channel capacity 5,8 , 12 2 1 2 where Alice has the first qubit, Bob the second, and ρ ρ ρ = − S( ) Tr( log ) 2 is the von Neumann entropy. It is easy to confirm that the dense coding channel capacity of a Bell state is 2 bits. Note that, in this paper, C(ρ jk ) means the channel capacity when Alice has the j th qubit, Bob the k th qubit, and C(ρ 12 ) = C(ρ 21 ) for a pure state ρ 12 .
In the controlled dense coding scenario 15 , there is another party, Charlie, who plays the role as a controller, and the three parties share a three-qubit state ρ 123 . To maximize the channel capacity between Alice and Bob, Charlie measures his qubit on an optimal basis that maximizes the average dense coding channel capacity and broadcasts the measurement outcome to Alice and Bob. After receiving the measurement outcome, Alice and Bob perform the dense coding scheme. We define the controlled dense coding channel capacity that Alice and Bob can achieve using this protocol as, where Charlie, Alice, and Bob have the j th, k th, and l th qubit, respectively (j, k and l are distinct numbers in {1, 2, 3}). The maximum is taken over all 2 × 2 unitary matrices U, which correspond to Charlie's choice of measurement basis. ρ ρ = Tr ( ) j kl 123 and ρ kl i is the quantum state that Alice and Bob share when Charlie's measurement On the other hand, we now consider the channel capacity that Alice and Bob achieve without Charlie's assistance. In this case, the quantum state that Alice and Bob share is the reduced density operator of Alice and Bob, ρ kl = Tr j (ρ 123 ) where Alice and Bob have the k th and l th qubit, respectively. Thus, the channel capacity without Charlie's assistance is given by C(ρ kl ), which we will denote by C j (ρ kl ) to specify Charlie's qubit. Finally, we define the CP of Charlie as the difference between the channel capacities with and without Charlie's assistance, jkl CD jkl j kl 123 and also define the minimal control power (MCP) where the minimum is taken over all possible permutation of {j, k, l}. MCP is defined in the same way as for controlled teleportation 20 by replacing teleportation fidelity with channel capacity.
Examples. Before we investigate the properties of MCP, let us calculate the MCPs of certain three-qubit states such as the extended GHZ states and generalized W states 23 .
Extended GHZ states. Let ψ eGHZ 123 be a state defined by To calculate CP, we need to obtain the channel capacities with and without Charlie's assistance for each permutation of {j, k, l}. The channel capacities with assistance can be obtained as follows: Thus, the CPs are given by , MCP of the extended GHZ states is and λ = 1/2 3 2 , i.e., when the state is the standard GHZ state, + ( 000 111 )/ 2 . In addition, MCP is zero if and only if λ 1 = 0 or λ 3 = 0, i.e., the state is biseparable or fully separable 24,25 . These facts imply that MCP is closely related to the genuine tripartite entanglement of pure states. To investigate the meaning of the result in terms of the tripartite entanglement, let us consider the three-tangle τ 25,26 which is defined for a pure three-qubit state ψ 123 as 123 , and  is Wootters' concurrence 27,28 . The three-tangle for the extended GHZ states is calculated as Thus, MCP is a monotonically increasing function of the three-tangle τ, Indeed, the three-tangle τ is known as a value that quantifies genuine tripartite entanglement. Therefore, MCP can also be used to quantify the genuine tripartite entanglement for extended GHZ states. Note that MCP of the controlled teleportation is also a monotonically increasing function of the three-tangle 20 .
In the special case that λ 2 = 0, the state ψ eGHZ is reduced to generalized GHZ states, gGHZ 123 1 3 Using , we can simplify channel capacities with Charlie's assistance in equations (6) and (7) as CD jkl 1 2 and channel capacities without Charlie's assistance in equations (8), (9) and (10) as j kl for all j, k, l. Thus, MCP of generalized GHZ states can be obtained as gGHZ 1 2 The difference between channel capacities with and without Charlie's assistance and MCP for generalized GHZ states are shown in Fig. 1. The figure shows how much channel capacity is affected by controller's assistance for generalized GHZ states.
For another special case (λ 1 = 1/√2), The MCP of generalized W states is calculated in the same way as before. First, the channel capacity with assistance is In contrast to the case of extended GHZ states, the measurement basis that maximizes the channel capacity is | { 0 , 1 }, regardless of the values of λ i 's. The channel capacity without assistance is After simplification, the CPs can be obtained as and, thus, MCP can be written as It is easily checked that MCP of a generalized W state is zero if and only if one λ i is zero, i.e., the state is biseparable or fully separable. In addition, we prove that the standard W state, + + ( 100 010 001 )/ 3 , has maximal MCP among the generalized W states as follows. , that is Although MCP for a given extended W state can be calculated easily, it is not easy to obtain an analytic expression for arbitrary extended W states. Nevertheless, we generated 10 5 extended W states randomly and calculated their MCPs, and no extended W state has been found that has a larger MCP than that of the standard W state, 2/3, which is shown in Fig. 2. Furthermore, we can see from Fig. 2(a) that for a given λ 0 2 , no extended W state has been found that has a larger MCP than that of the extended W state with λ 1 = λ 2 = λ 3 . We also conclude that MCP of the extended W states with λ 1 = λ 2 = λ 3 is a monotonically decreasing function of λ 0 2 . In addition, we also plot the same data against λ 1 2 , which is shown in Fig. 2(b). From the figure, we can also see that for a given λ 1 2 , no extended W state has been found that has a larger MCP than that of the extended W state with λ 2 = λ 3 and λ 0 = 0.

Properties of minimal control power.
In this section, we investigate properties of MCP in terms of genuine tripartite entanglement. Before we start to introduce the properties of MCP, let us rewrite equation (7) as , which corresponds to maximization over all Charlie's possible measurement basis 30 . Thus, CP also can be written as k l , which is also known as a negative quantum conditional entropy 31,32 . Using the properties of the von Neumann entropy and convexity of coherent information, we can prove the following proposition.

Proposition 2.
For general three-qubit states ρ 123 , which can be mixed or pure,

123
Before we start to prove the proposition, we review the inequalities for the von Neumann entropy of the mixture of quantum states ρ i Proof. Let us prove the lower bound first. It can be shown that CP of ρ 123 is non-negative by using the convexity of the coherent information 31, 32 , Thus, MCP is also non-negative, P(ρ 123 ) ≥ 0. For the upper bound, using equation (35), we obtain for CP that  . If ρ 123 is biseparable or fully separable, it is clear that when Charlie has the separated qubit, CP is zero, and thus MCP is zero. Let us now assume that P(ρ 123 ) = 0. Thus, there exists {j, k, l} such that P jkl (ρ 123 ) = 0. We set {j, k, l} = {3, 1, 2} without loss of generality. Equation (36) implies that for any ensemble because the maximum is zero. Let us write the state in a Schmidt decomposition form ψ µψ 12 3 , and let us assume that μ i ≠ 0 because ψ 123 is trivially biseparable or fully separable if μ 0 = 0 or μ 1 = 0. Then, it is easy to obtain that ρ Proof. We have already shown that MCP of the standard GHZ state is 1 from equation (13). Now, let us assume that MCP of a pure three-qubit state ρ 123 is 1 so that the CPs of ρ 123 for any {j,k,l} are 1. Note that since ρ = S( ) 0 kl i for pure states ρ 123 , CP can be written as To prove that a pure state ρ 123 that attains the upper bound should be the standard GHZ state, we need to prove that the measurement basis of Charlie that maximizes P jkl also maximizes P jkl and vice versa.
Since the eigen-decomposition of ρ 1 and ρ 2 is unique, we require (i) λ µ , and = i i . Thus, we can only require condition (ii), which gives us that ψ 0 and ψ 1 are two orthogonal Bell states. Hence, ψ 123 is the standard GHZ state up to a local unitary operation. ☐

Discussion
We have considered controlled dense coding and defined its CP and MCP. We have also calculated MCPs for several important pure three-qubit states such as the extended GHZ states and generalized W states. We found that MCP of the extended GHZ states is a monotonically increasing function of the three-tangle of the states, which quantifies genuine tripartite entanglement. We also found that the standard GHZ state uniquely achieves the maximal value of MCP among the extended GHZ states, so does the standard W state among the generalized W states. Although the extended W states have a genuine tripartite entanglement, the three-tangle of the extended W states vanishes 25 . Based on the operational meaning of MCP, the three-party entanglement of the extended W states can be witnessed by MCP as we have discussed in this paper. Thus, MCP can be used to quantify the three-party entanglement of the extended W states.
We also found the lower and upper bounds of MCP for general three-qubit states. In addition, we proved that for pure states ρ 123 , the equality of the lower bound holds if and only if the three-qubit state is biseparable or fully separable and that the equality of the upper bound holds if and only if the three-qubit state is the standard GHZ state. These properties imply that not only does MCP have an operational meaning in the controlled dense coding scenario but also that it can capture the genuine tripartite entanglement of three-qubit pure states. We remark that it is possible for a separable mixed three-qubit state to attain the maximal value of MCP, for example, ρ = | 〉〈 | + | 〉〈 | + | 〉〈 | + | 〉〈 | . Furthermore, it also means that there exists a mixed state whose MCP is not zero even though the state is fully separable. The interpretation is that both classical and quantum correlations of mixed three-qubit states ρ 123 are captured by MCP. Thus, in future work, it would be worth analyzing the relation between MCP and genuine tripartite quantum and classical correlations.
Even though we have analyzed discrete variable controlled dense coding only, it is worth mentioning generalization to continuous variable (CV) systems. In fact, CV dense coding and CV controlled dense coding have been studied 16,[33][34][35] . In order to perform CV dense coding, a two-mode squeezed vacuum is shared by Alice and Bob, and Alice encodes CV information by displacing her beam and sends it to Bob, followed by Bob's joint measurement of quadrature variables. For controlled dense coding, a GHZ-like state of CV is shared among Alice, Bob and Charlie, and Charlie performs homodyne detection on his beam. The result of the homodyne measurement is then sent to Alice, and Alice and Bob perform CV dense coding using Charlie's measurement result. In fact, it was shown that channel capacities with and without Charlie's assistance are different 35 . At a first glance, it seems possible to define the CP and MCP of CV controlled dense coding in a similar manner. However, channel capacity of CV dense coding and that of discrete variable have some differences. The channel capacity of CV dense coding requires certain constraints to prevent it from diverging, which are not required for discrete variable dense coding. In addition, the channel capacities that we have defined for discrete variable controlled dense coding are the optimal ones for given states so that it can be interpreted as an intrinsic property of the state, whereas optimal channel capacities of CV states are not known in general. Thus, in order to define CP and MCP properly and generalize our results to CV systems, it is required to find a general expression of the optimal channel capacity for a given CV state. It would be an interesting future work to generalize the CP and MCP of discrete variable controlled dense coding to CV controlled dense coding.