Controlled Remote State Preparation via General Pure Three-Qubit State

The protocols for controlled remote state preparation of a single qubit and a general two-qubit state are presented in this paper. The general pure three-qubit states are chosen as shared quantum channel, which are not LOCC equivalent to the mostly used GHZ-state. It is the first time to introduce general pure three-qubit states to complete remote state preparation. The probability of successful preparation is presented. Moreover, in some special cases, the successful probability could reach unit.


Introduction
Quantum teleportation (QT for short) is the first quantum information processing protocol presented by Bennett et al. [1] to achieve the transmission of information contained in quantum state determinately. Many theoretical schemes have been proposed later [2,3,4,5,6]. It has also been realized experimentally [7,8,9,10,11,12,13,14]. Latter, to save resource needed in the process of information transmission, Lo put forward a scheme for remote preparation of quantum state (RSP for short) [15]. Compared with QT, in RSP the sender does not own the particle itself but owns all the classical information of the state he or she wants to prepare for the receiver, who is located separately from the sender. The resource consumption is reduced greatly in RSP, as the sender do not need to prepare the state beforehand. The RSP has already attracted many attentions. A number of RSP protocols were presented, such as RSP with or without oblivious conditions, optimal RSP, RSP using noisy channel, low-entanglement RSP, continuous variable RSP and so on [16,17,18,19,20,21,22,23,24,25,26]. Experimental realization was also proved [27,28].
In RSP protocols, all the classical information is distributed to one sender, which may lead to information leakage if the sender is not honest. In order to improve the security of remote state preparation, controllers are introduced, which is the so called controlled remote state preparation (CRSP for short), and it has drawn the attention of many researchers. In contrast to the usual RSP, the CRSP needs to incorporate a controller. The information could be transmitted if and only if both the sender and receiver cooperate with the controller or supervisor. CRSP for an arbitrary qubit has been presented in a network via many agents [29]. A two-qubit state CRSP with multi-controllers using two non-maximally GHZ states as shared channel is shown in [30]. CRSP with two receivers via asymmetric channel [31], using POVM are presented [32,33]. The five-qubit Brown state as quantum channel to realize the CRSP of three-qubit state is elaborated in [34]. Most of the existing schemes chose to use the GHZ-type state, W-type state, Bell state or the composite of these states as the shared quantum channel. However in this paper, we choose the general pure threequbit state as quantum channel, which is not LOCC equivalent to the GHZ state. And for some special cases, the probability for successful CRSP can reach unit.
In [35], the authors proved that for any pure three-qubit state, the existence of local base, which allows one to express a pure three-qubit state in a unique form using a set of five orthogonal state. It is the called generalised Schmidt-Decomposition for three-qubit state. Using the generalised Schmidt-Decomposition, Gao et al. [36] proposed a controlled teleportation protocol for an unknown qubit and gave analytic expressions for the maximal successful probabilities. They also gave an explicit expression for the pure three-qubit state with unit probability of controlled teleportation [36]. Motivated by the ideas of the two papers, we try to investigate the controlled remote state preparation using the general pure three-qubit states and their generalised Schmidt-Decomposition.
The paper is arranged as follows. In Sec. 2, the CRSP for an arbitrary qubit is elucidated in detail. We find that the successful probability is the same as that of controlled teleportation for qubits with real coefficients. In Sec. 3, the CRSP for a general two-qubit state is expounded. For two-qubit state with four real coefficients. The corresponding successful probability is the same as that of controlled teleportation of a qubit. In Sec. 4, we conclude the paper.

CRSP for an arbitrary qubit
Suppose that three separated parties Alice, Bob and Charlie share a general pure three-qubit particle |Φ cab , the particle a belongs to Alice, b to Bob and c to Charlie, respectively. The distribution of the three particles are sketched in Fig.1. In figure  1, the small circles represent the particles, the solid line between two circles means that the corresponding two particles are related to each other by quantum correlation. According to [35], the general pure three qubit state has a unique generalised Schmidt-Decomposition in the form where a i ≥ 0 for i = 0, · · · , 4, 0 ≤ µ ≤ π, ∑ 4 i=0 a 2 i = 1. The a i and µ in Eq.(1) are decided uniquely with respect to a chosen general pure three qubit state. Now Alice wants to send the information of a general qubit to the remote receiver Bob under the control of Charlie. Alice possesses the classical information of this qubit, i.e. the information of α and β , but does not have the particle itself. Next, we make three steps to complete the CRSP for |ϕ .
Step 1 The controller Charlie firstly makes a single qubit measurement under the base |ε 0 c = cos where θ ∈ [0, π], η ∈ [0, 2π]. The choice of θ and η could be flexible according to the need of the controller. If θ = π and η = 0, |ε 0 c and |ε 1 c will be the |± base. Then Charlie broadcasts his measurement outcomes publicly to Alice and Bob using one classical bit. Using Eq.(2), the quantum channel can be rewritten as where p 0 = sin 2 θ 2 + a 2 0 cos θ + a 0 a 1 cos(µ − η) sin θ , If the result of Charlie's measurement is 0, the whole system collapses to |Ω 0 ab with probability p 0 while collapses to |Ω 1 ab with probability p 1 for the result 1. To ensure that the particle c entangles with the whole system, we assume that a 0 > 0 and a 2 , a 3 , a 4 are not equal to 0 at the same time. This is equivalent to p 0 > 0 and p 1 > 0 at the same time. Note that Step 1 is actually similar to that of controlled teleportation in [36]. We arrange it here to keep the integrity of the paper. More detailed calculation can be found in [36].
Step 2 Without loss of generality, we assume that the result of Charlie's measurement is 0. Then the whole system collapse to |Ω 0 ab . Using the Schmidt-Decomposition of two-qubit system, there exists bases {|0 , |1 } a and {|0 , |1 } b for particle a and b respectively, such that |Ω 0 ab can be expressed as where [36]. On receiving the result of Charlie's measurement, the sender Alice prepares a projective measurement utilizing the classical information of |ϕ in the following form: Then |Ω 0 ab could be reexpressed as Next we first discuss the case for real coefficients, i.e. α, β are real. Then Eq.(6) will be Alice measures her qubit under base {|µ 0 , |µ 1 } a and gets the outcome 0 and 1 with probability λ 00 α 2 + λ 01 β 2 and λ 00 β 2 + λ 01 α 2 respectively. And Alice sends her measurement result to Bob by 1 classical bit. The receiver Bob's system will collapse to respectively.
Step 3 We assume that Alice's measurement result is 0. Now according to Charlie and Alice's result, Bob wants to recovery the state |ϕ on his side. Bob needs to introduce an auxiliary particle in initial state |0 b , then he makes a unitary operation U 0 bb on his particle b and the auxiliary particle b , and his state changes to |ω 0 bb , where After the unitary operation, Bob makes a measurement on his auxiliary particle b under the base {|0 , |1 } b . The probability for Bob to get measurement result 0 is λ 00 /(λ 00 α 2 + λ 01 β 2 ), and he can recovery state |ϕ successfully. But if the result is 1, the scheme fails.
Similarly, if Alice's measurement result is 1, Bob also introduces an auxiliary particle in initial state |0 b . But the unitary operation is U 1 bb , and the system after the unitary operation is |ω 1 bb , where The probability for Bob to successfully reconstruct the state |ϕ is λ 00 /(λ 00 β 2 + λ 01 α 2 ).
Then continuing to use the last 2 steps as those in Charlies measurement result is 0, we can get that the successful probability for Bob to produce the desired state is 2p 1 λ 10 . As a result, for the real case, Alice can prepare the qubit |ϕ at Bob's position under the control of Charlie with probability 2(p 0 λ 00 + p 1 λ 10 ), which is the same as that of controlled teleportation in [36]. But the consumption of classical bits is reduced to 2 cbits for the whole process.
Next we discuss the case for complex coefficients.
According to the discussion of [36], the maximally probability for controlled teleportation will reach unit if and only if the shared channel is As for the controlled remote state preparation for a qubit using the above channel, the successful probability can also reach one for the real case, and 1/2 for the complex case.

CRSP for a two-qubit state
In the CRSP for a two-qubit state, there are also three parties Alice, Bob and Charlie. They share a quantum channel which is the composite of |Φ cab and the Bell state, the distribution of particles in the shared quantum channel is displayed in Fig.2, the meaning of symbols is the same as in Fig.1.
the particle c belongs to Charlie, a, a to Alice and b, b to Bob. Now the sender Alice possesses the classical information of a general two qubit state |ϕ , Step 1 This step is the same as that of Step 1 in Section 2. Charlie makes a projective measurement {|ε 0 c , |ε 1 c } on his particle c, and gets the measurement result 0 and 1 with probability p 0 and p 1 respectively. The whole system collapses to |Ω 0 ab |φ + a b and |Ω 1 ab |φ + a b respectively. He broadcast his measurement result using 1 cbits.
Step 3 Assume that the measurement result of Alice is 0 in Step 2. Then according to the result, Bob introduces an auxiliary particle b a in the initial state |0 b a , and makes unitary operation The state after Bob performing the unitary operation is .

Conclusions
In this paper, protocols for controlled remote state preparation are presented both for a single qubit and two-qubit state. We utilize the general pure three qubit states as the shared quantum channels, which are not LOCC equivalent to the GHZ state. We discuss protocols for both states with real and complex coefficients, and find that the general pure three-qubit states can help to complete CRSP probabilistically. More than that, in some spacial cases, the CRSP can be achieved with unit probability, which are deterministic CRSP protocols. This overcomes the limitation that most of the existing quantum communication protocols are completed with GHZ-, W-or Bell states, or the composition of these states. Moreover, due to the involvement of controller and multi-partities, this work may have potential application in controlled quantum communication, quantum network communication and distributed computation.