It is well known that an unknown quantum state can not be perfectly copied because of the quantum no-cloning theorem [1,2]. Since the seminal work of Bužek and Hillery [
All the previous RQIC protocols were focused on the reverse process of the optimal universal telecloning. In this paper, we investigate the reverse process of ancillafree phase-covariant telecloning. Particularly, we present a scheme for implementing 2→1 RQIC, i.e., the reversal of the optimal 1→2 ancilla-free phase-covariant telecloning (AFPCT) for qubits. The entangled channel used in our RQIC protocol is the same as that used in the optimal 1→2 AFPCT [
Before describing our RQIC protocol, we briefly summarize the forward process, the optimal 1→2 AFPCT [
from a distributor (Alice) to two spatially separated receivers (Bob and Charlie) with only LOCC. Here 
represents the computational basis for a qubit and δ is an unknown phase. The entangled channel is a three-qubit asymmetric W state [
where qubit
is the input port of the distributor Alice, and qubits
and
are the output ports for the clones held, respectively, by the two receivers Bob and Charlie (throughout the paper, if necessary, the subscripts of the kets or the operators denote the qubit index). Here we have adopted the notation
. The telecloning can be accomplished by the following simple procedure: 1) Alice performs a Bell-state measurement on qubits A and
, obtaining one of four Bell states, and informs Bob and Charlie of the outcome; 2) depending on Alice’s measurement outcome, Bob and Charlie perform appropriate local unitary operations on the qubits they hold, obtaining the clones of
with the optimal fidelity
[27,28]. The collective output state of qubits 
and
is the entangled state
We now describe our RQIC protocol for implementing the reversal of the aforementioned telecloning. After telecloning operations, the initial single-qubit (A) quantum information is remotely distributed into two spatially separated qubits (
and
), represented by the collective quantum state in Equation (3). The ownership of particles
and
is the same as before; i.e., they are still held by Bob and Charlie, respectively. The RQIC is aim to concentrate the information initially distributed in two-qubit cloning state of Equation (3) back to a remote particle D (held by Diana) with only LOCC: 
We also consider employing the following tripartite asymmetric W entangled state as the quantum channel:
where particles B, C, and D belong to Bob, Charlie, and Diana, respectively. The RQIC task can be accomplished by three steps: (S1) Bob and Charlie perform Bell-state measurements on their own particles, respectively; (S2) Each of them tells Diana the measurement outcome by sending 2 bits of classical information; (S3) Diana performs a conditional local operation on particle D.
The detailed explanation for the above procedure is as follows. According to Equations (3) and (4), the state of the whole system (qubits {
,
,B, C, D}) is given by
With the Bell-basis states
The total quantum state 
can be expanded as
It can be seen from Equation (7) that when the combinational Bell-state measurement outcome is in the set
the particle D will be projected in the state
, meaning the success of RQIC. If the double Bell-state measurement outcome belongs to the following set
, the particle D can also be prepared in the state
, by suitable local unitary transformations (see
on the two pairs of qubits 
and
. 
denote conventionally the Pauli operators and I is the identity operator.
As to the physical realization of the above RQIC scheme, one mainly needs to consider three points: 1) preparation of the entangled channel, i.e., the asymmetric W state of Equation (4); 2) realization of the optimal 1→ 2 ancilla-free telecloning (or cloning) of the phase-covariant state 
in Equation (1); 3) implementation of the Bell-state measurement. Recently, some schemes for generating the asymmetric W state of Equation (4) have been presented in different physical systems [29-33]. Several physical schemes for realizing the optimal 1→2 ancilla-free phase-covariant telecloning or cloning have also been proposed [33,34]. Based on the current techniques [35,36], these schemes are experimentally achievable. The Bell-state measurement can be well realized for both atomic and photonic qubits [37,38]. Very recently, we have also proposed a near deterministic scheme [
In conclusion, we have presented a new RQIC scheme for implementing the reversal of the optimal 1→2 AFPCT, in contrast to the previous RQIC schemes for implementing the reversal of the optimal universal telecloning. The entangled channel used in our protocol is an experimentally realizable three-qubit W state. Such a RQIC protocol is expected to find useful applications in the field of modern multiparty quantum-information processing. In addition, the idea may open another perspective for the applications of the W states.
There are two open questions that deserve further investigating in the future. 1) Whether the reverse process of the optimal 1→2 AFPCT could be implemented deterministically? 2) Whether the reverse process of 1→n AFPCT could be realized?