Optimal joint remote state preparation in the presence of various types of noises

A main obstacle faced by any quantum information processing protocol is the noise that degrades the desired coherence/entanglement. In this work we study by means of Kraus operators the effect of four typical types of noises on the quality of joint remote state preparation of a single-qubit state using a three-qubit Greenberger–Horne–Zeilinger-type state as the initial quantum channel. Assuming that two of the three involved qubits independently suffer a type of noise, we derive analytical expressions not only for the optimal averaged fidelities but also for the boundaries in phase space of the domains in which the joint remote state preparation protocol outperforms the classical one. Detailed discussion is given for each of the total 16 noisy scenarios. We also provide physical interpretation for the obtained results and outline possible future topics.


Introduction
Joint remote state preparation (JRSP) is a global protocol [1][2][3] in which a group of M people, called the preparers, remotely cooperate to provide a quantum state ψ for another distant person, called the receiver, in such a way so as to meet two following prerequisite conditions: (i) only local operations and classical communication are allowed and (ii) any individual one or any subgroup of the preparers cannot infer the full information encoded in the state ψ . The condition (i) is ensured by sharing among the preparers and the receiver a proper multipartite entangled resource. As for the condition (ii), the full classical information of ψ is secretly divided into M pieces, each of which is independently given to a preparer. Such a secret information splitting makes JRSP secure compared to remote state preparation (RSP) protocols (see e.g. [4]) in which there is only one preparer who catches the entire information about ψ . Technically, JRSP and RSP just require single-qubit von Neumann measurements so they are less demanding than quantum teleportation [5] that needs collective Bell-measurements. Since its introduction a great deal of papers on JRSP has been published in diverse aspects [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]. In its early development, only probabilistic protocols were devised [1-3, 6-10, 13-16, 18, 21-24]. Later it turns out that JRSP can be made deterministic by adopting adaptive measurement strategies [11,17,19,20] (i.e. the preparers carry out their measurements in sequence with the outcome of an earlier measuring preparer being feed forwarded to a next preparer who will make use of it to choose the right basis for his/her own measurement). JRSP is also approached to from an experimental architecture point of view [25] and, recently, a scheme for realizing JRSP of photonic states with linearoptics devices has been proposed [26].
Like other quantum tasks, JRSP suffers a serious problem under the name decoherence caused by unavoidable interactions with surrounding noisy environments during the preliminary stage of entanglement distribution that make an intended quantum channel pure state become mixed one with lesser degree of entanglement. If the source of quantum resources is generous, special procedures can be applied to distill a desired state from an ensemble of decohered states, provided that fidelity of the decohered states with respect to the desired state is not too low (namely, not smaller than 1/2) [27][28][29][30][31]. However, distillation procedures consume a heavy overheads in both quantum resource/technology and time. A possible way out of the situation is to directly employ the decohered state. In this perspective, studying the effect of noises is truly necessary to optimize the performance of a given quantum protocol. A few authors already devoted their interest to noisy JRSP using different kinds of shared quantum channels through Lindblad master equations within the framework of Markov-Born approximations [32][33][34][35]. In this work, motivated by the study in [36] for noisy quantum teleportation, we shall resort to the apparatus of Kraus operators to investigate similar issues but with respect to JRSP. A particular new result compared with previous works is that besides analytical expressions for the optimal averaged fidelities we also derive expressions in terms of noise strengths for the boundaries of the domains in which quality of the JRSP protocol is better than that achievable by any classical means. Note that very recently the authors of [37] have used the same mathematical apparatus to deal with the same topic but with simpler noisy scenarios, so their results are contained in ours as particular cases.
We structure our paper in four sections. In the next section, section 2, we shall for clarity present the general formalism of the = M 2 JRSP of a single-qubit state in density matrix language. Section 3 will analyze in detail various scenarios of four standard noise types acting on two of the three qubits of concern. Each scenario has its own consequences which will be elucidated in due places in section 3. The final conclusion section, section 4, will summarize our results with physical interpretations and briefly list on what could be done subsequently.

General formalism
Consider a case when there are two preparers (Alice and Bob) and what to be remotely prepared is a single-qubit state of the most general form with [ ] π ϑ ∈ 0, and [ ] ϕ π ∈ 0, 2 . Alice knows only ϑ, while Bob knows only ϕ, so neither of them knows ψ . The quantum channel shared beforehand among the two preparers and the receiver (Charlie) is in general an entangled mixed state ρ 123 , with qubit 1 (2, 3) held by Alice (Bob, Charlie). The joint remote preparation of ψ begins with Alice who measures qubit 1 in the basis with ( ) ϑ U a specific unitary operator depending on ϑ and on the initial condition. If her measurement outcome is k (i.e. u k 1 is found), qubit 1 is disentangled from ρ 123 but Bob's and Charlie's qubits remain entangled with their state being in the form is a specific unitary operator depending not only on ϕ but also on k. If Bob finds ( ) v l k 2 (i.e. his outcome is l which should also be publicly announced), state ( ) ρ k 23 becomes separable: is the probability of Bob's obtaining the outcome l. Finally, after collecting the measurement outcomes k and l announced by Alice and Bob, Charlie applies on ( ) ρ k 23 a suitable unitary operator ( ) R kl to obtain the state kl kl kl kl 3 (8) whose fidelity in comparison with the desired state ψ in equation (1) is determined by Averaging over the four possible measurement outcomes yields Generally, F depends on the quantum channel and the state to-be-prepared, so to have a state-independent fidelity we further average over all the possible parameters of the input state. Assuming a uniform distribution, the state-independent averaged fidelity F can be calculated following the formula whose dependence on the quantum channel however remains.

Effects of noises
It is ideal if there are no noises and the initial quantum channel is a pure maximally entangled state which is not unique but may be Einstein-Podolski-Rosen (EPR) states [2, 3, 6, 11-13, 17, 20], Greenberger-Horne-Zeilinger (GHZ) state [1-3, 6-8, 15, 19], W state [9,10,14] or others [16,18,[21][22][23][24]. In case it is the GHZ state , the unitary operators in equations (2) and (5) are chosen as and with θ introduced as a freely controlling parameter to optimize the JRSP performance. The effect of noises can conveniently be accounted for by means of superoperators whose action on a density matrix yields again a legitimate density matrix. In the operator-sum representation the mentioned superoperator is given in terms of Kraus operators [39] which adequately model a specific type of noise. Four typical noise types which are often encountered in reality are bit-flip (B), phase-flip (P), amplitude-damping (A) and depolarizing (D) (the physical meaning of a noise type can be found in [39,40]). Suppose that each of the qubits 1, 2, 3 independently experiences a type of noise. Subjected to such noise types, the initial pure quantum channel state ( ) θ Q 123 in equation (14) becomes a mixed one, ρ 123 , which can be represented in terms of Kraus operators as where α α K p j 1 ( ) and α p 1 are the jth Kraus operator and the noise strength for qubit 1 under the action of the noise type { } α ∈ B P A D , , , , while α N is the number of α-type noise Kraus operators. The noise strength is a parameterized quantity which is proportional to the time the noise is acting on the qubit or the distance the qubit has to travel along in the noisy environment. If the time/distance is zero the noise strength is zero. Infinity of the time/distance is meant by unit noise strength. Hence, α p 1 satisfies the conditions ⩽ ⩽ α p 0 1 1 and is in essence a probability. Similar explanations hold for ( ) Let Bob be capable of producing the initial quantum channel ( ) θ Q 123 at his wellequipped laboratory. After that, he keeps qubit 2 with himself in a noise-free storage, but sends qubit 1 (3) to Alice (Charlie) via α-type (γ-type) noisy environment. Hence, the resulting channel reduces to and Calculations based on the above formulae for the Kraus operators and those given in the general formalism section give (24) At this moment the parameter θ introduced in equation (14) will play its role. Since ⩽ ⩽ , it is trivial from equations (21), (22) and (24) that the optimal values of θ at which F BB , F BP and F BD become maximal are With so chosen values of θ γ B opt we have straightforwardly derived the optimal averaged fidelities γ F B opt from equations (21)- (24). Their analytical expressions read and The dependences of 3 are plotted in figure 1. As a rule, the JRSP protocol is useless if the optimal averaged fidelities γ F B opt is equal to or smaller than 2/3, the best classically achievable fidelity value [41,42].
To visualize the domain of p B 1 and γ p 3 within which the JRSP protocol remains useful we display in figure 2 the corresponding density plots in phase spaces. As quickly followed from figures 1(a) and 2(a), F BB opt decreases with increasing p B 1  . Moreover, for a given , at which F BP opt drops to 2/3, and then the protocol ceases to work for any further increase in p B 1 . On the other hand, for a given p B 1 the value of F BP opt decreases as p P 3 increases from zero, which is usual. This tendency continues until , at and immediately beyond which the protocol 'dies' (i.e. out of service). Interestingly, however, as p P 3 increases further to reach a value larger than ( )/ + p 1 2 B 1 the protocol 'revives' (i.e. back to service) with F BP opt increasing with p P 3 , which seems unusual since it would mean that 'more noise better quality' (i.e. larger p P 3 leads to larger F BP opt ). Such sudden 'death' and sudden 'birth' of service of the quantum JRSP protocol for F BP opt are . This says that non-maximally entangled channel ( ( / ) ) θ π ≠ Q 4 123 outperforms maximally entangled channel ( ( / ) ) θ π = Q 4 123 or 'less entanglement better quality'. As for figures 1(d) and 2(d), the useful domain in phase space is narrower than that in figures 1(c) and 2(c), with the border determined by Next, consider α = P and { } γ ∈ B P A D , , , . In this scenario the values of θ that make the JRSP protocol optimal can be found to be Note that use of the above value of θ PA opt reduces the entanglement degree of the initial quantum channel ( ) θ Q 123 in equation (14). The optimal averaged fidelities are derived in the form and The dependences of γ F P opt on p P 1 and γ p 3 ( ) γ = B P A D , , , are plotted in figure 3 and the phase-space useful domains are shown in figure 4. The border between the useful and useless domains for F PB opt in figure 4(a)  the JRSP protocol becomes more and more efficient with increasing phase-flip noise. When qubit 1 is affected by phase-flip noise and qubit 3 (a) also by phase-flip noise or (b) by amplitude-damping noise, the protocol is better than the  , as is evident from figures 4(b) and (c). When qubit 1 is affected by phaseflip noise and qubit 3 by depolarizing noise, the phase-space diagram shown in figure 4(d) is similar to that in figure 4(a), but with the border described by a different equation, , so that of p D 3 cannot be equal to or greater than 2/3 and for a given With any of the above θ γ A opt the initial quantum channel ( ) θ Q 123 is non-maximally entangled but the corresponding averaged fidelities are maximal (45) and Note that although the precise calculations yield different expressions for the averaged fidelities (i.e.
The optimal averaged fidelities are derived in the form and The optimal averaged fidelity F DB opt ( F DP opt , F DA opt and F DD opt ) as a function of p D 1 and ( P P P , . The phase-space diagrams in figure 8 tell that generally the allowed values of p D 1 and γ p 3 are more limited in this scenario compared with those previously considered.

Conclusion
We have drawn a small stroke in the big picture of decoherence by analyzing how noises affect the quality of JRSP using a controllable GHZ-type state as the initial quantum  channel. The types of noises we have considered are bit-flip, phase-flip, amplitude-damping and depolarizing noise which are usually met in realistic situations. The action of a noise type on a qubit is modeled by a superoperator in terms of the operator-sum of Kraus operators. We assumed that two of the three concerned qubits are acted on by an independent noise and derived explicit dependences of optimal averaged fidelities of JRSP in all possible 16 scenarios. In each scenario we also figured out the phase space diagrams allowing to specify the range of noise strengths in which the quantum JRSP protocol is better than the best classical one. Beside the usually expected property that quality gets worse with increasing noise or/ and decreasing degree of entanglement, we encountered two 'unusual' things. The first one is 'more noise better quality' and the second one is 'less entanglement better quality'. These things also arose in noisy quantum teleportation [36] as a direct result of mathematical calculations. Here we provide physical interpretations. The point is that quality is decided by the working quantum channel state. In the presence of noises the working state is the decohered one but not the initial one. More concretely, in the context of our JRSP protocol, the working state is ( ) ρ θ αγ 123 in equation (16) but not the initial one ( ) θ Q 123 in equation (14). Because the ideal state ( / ) θ π = Q 4 123 is maximally entangled, it is natural that the closer increases with the noise strength when it is larger than 1/2. Hence, the 'more noise better quality' is physically translated to 'larger entanglement better quality' (note, here it is entanglement of the working channel is maximal at / θ π ≠ 4. That is, the initial state ( / ) θ π ≠ Q 4 123 is non-maximally entangled, but the entanglement degree of the working state ( / ) ρ θ π ≠ αγ 4 123 is larger than the entanglement degree of ( / ) ρ θ π = αγ 4 123 . So, the 'less entanglement better quality' also implies 'larger entanglement better quality' (note, in the former italic sentence entanglement means entanglement of the initial quantum channel but in the latter italic sentence entanglement means entanglement of the decohered quantum channel which is the working one). Therefore, in light of our interpretations the two 'unusual' things mentioned above turn out to be usual: both in fact reflect the truth that a working (not initial) quantum channel with a larger degree of entanglement results in a better quality of JRSP protocol. For a comprehensive study one can extend this work to the case when all the three qubits are subjected to noises or to consider other kinds of the initial quantum channels or analyze other ways of noise action (e.g. collective action of different types of noises, non-Markovian noise, etc). Noisy JRSP and noisy controlled JRSP of multi-qubit states are also worth investigating. These topics may be dealt with in future.