NMR Experimental Demonstration of Probabilistic Quantum Cloning
The method of quantum cloning is divided into two main categories: approximate and probabilistic quantum cloning. The former method is used to approximate an unknown quantum state deterministically, and the latter can be used to faithfully copy the state probabilistically. So far, many approximate cloning machines have been experimentally demonstrated, but probabilistic cloning remains an experimental challenge, as it requires more complicated networks and a higher level of precision control. In this work, we designed an efficient quantum network with a limited amount of resources, and performed the first experimental demonstration of probabilistic quantum cloning in an NMR quantum computer. In our experiment, the optimal cloning efficiency proposed by Duan and Guo [Phys. Rev. Lett. 80, 4999 (1998)] is achieved.
pacs:03.67.-a, 03.67.Dd, 76.60.-k
The no-cloning theorem states that an arbitrary quantum state cannot be cloned perfectly Wootters299 (). This is a direct consequence of the linearity of quantum mechanics, and constitutes one of the most fundamental differences between the classical and quantum information theory. Remarkably, this property is also the crucial element for guaranteeing the security of many quantum key distribution protocols. Although quantum states can not be cloned faithfully, in a seminal paper, Buek and Hillery proposed an ”approximate cloning machine” that can produce two identical copies approximately close to the original one Buzek54 (). On the other hand, quantum cloning machines are of significant importance in quantum cryptography as they provide the optimal eavesdropping technique for a large class of attacks on many quantum key distribution protocols Bennett84 (); Niu60 () and have attracted a great deal of interest in further research Scarani77 (); Gisin79 (); Cerf85 (); Bruss62 (); Ariano64 (); Delgado98 (); Lamata101 (). Up to now, several approximate cloning machines have been experimentally demonstrated in optical systems Huang64 (); Antia296 (); Martini419 (); Fasel89 (); Zhao95 (); Bartuskova99 (); Nagli720 () and NMR systems Cummins88 (); Du94 (); Chen75 ().
Apart from the idea of an approximate cloning machine, an interesting alternative quantum cloning machine, probabilistic quantum cloning machine (PQCM), was proposed by Duan and Guo Duan80 (). They showed that states randomly chosen from a known set of states can be probabilistically cloned with perfect fidelity, if the states in the set are linearly independent. More specifically, if the states in the set are two non-orthogonal states and , the cloning fidelity will be 1 and the optimal cloning efficiency is given by .
There are many theoretical investigations on PQCM in literatures Pati83 (); Qiu39 (); Azuma72 (), and resource demanding experimental proposals exist Zhang61 (). However, to the best of our knowledge, no experiment has been reported until now. To make the experiment feasible, one should overcome two difficulties: (i) to minimize the quantum network complexity, and (ii) to achieve precise quantum control at a certain error threshold. In this work, we successfully solved these two problems and experimentally demonstrated the probabilistic quantum cloning machine with optimal cloning efficiency Duan80 ().
We start by introducing the experimental scheme to realize the optimal probabilistic quantum cloning. The quantum logic circuit for the probabilistic quantum cloning process is illustrated in Fig. 1. This is a three-qubit network, which contains one Hadamard gate, two controlled-NOT gates, and two controlled-rotation gates. In this network, qubit a is the probe qubit that indicates whether the cloning progress is successful. Qubit b is the to-be-cloned qubit, which is randomly chosen from the set , where
When , the states in the set are orthogonal. This simplification is reasonable because any pair of arbitrary states and can be transformed to the form of Eq. (1) via a unitary rotation. Qubit c is the cloning qubit and initially set to . After the unitary evolution, if qubit a is detected in state , the cloning process succeeds and we will obtain two perfect copies of the input state at qubit b and c; and if qubit a is detected in state , the cloning process fails.
The output state at the end of the quantum circuit can be written as:
where is a unitary operator for the quantum network shown in Fig. 1, the parameter is the cloning efficiency and is the normalized state of the composite system BC. The first term denotes the success of the cloning while the second term represents the failure. If we substitute the expressions of the unitary operator and the input states in Eq. (2), we obtain the cloning efficiency as:
which has been proven to be optimal Duan80 (); Pati83 (). Compared with the logic circuit of the cloning machine proposed in Ref.Zhang61 (), this scheme requires fewer quantum logic gates. This setup should be more robust in practice, as it is less affected by experimental imperfections, such as errors in radio-frequency pulses and decoherence.
Using a sample of Diethyl-fluoromalonate, the quantum circuit was implemented on a liquid-state NMR quantum-information processor. Three qubits are represented by the H, C and F nuclear spins. The molecular structure is shown in Fig.2(a), where the three nuclei used as qubits are marked by the oval. The natural Hamiltonian of three-qubits system in the rotating frame can be written as:
where represent Larmor frequencies, ’s are the coupling constants: Hz, Hz and Hz. The experiments were performed at room temperature using a Bruker Avance 400MHz NMR spectrometer equipped with a QXI probe with pulsed field gradient Zhang100 (); Peng103 ().
The system was first prepared in a pseudo pure state (PPS) , where describes the thermal polarization of the system and is the identity matrix, using the method of spatial averaging Cory120 (). From the state , we prepared the initial state through rotating qubit b by angle around the -axis.
The quantum circuit is realized by hard pulses and free evolution. The pulse sequence is depicted in Fig. 3. In principle, the readout procedure should be applied to each of the cloning qubits in the subsequent experiments at the end of the quantum circuit. In this experiment, a sample in natural abundance is used, i.e., only of the molecules had one C nuclear spin. To distinguish those molecules from the background molecules, we collect all signals from the cloning qubits through the C channel, by applying SWAP gates and measuring the C qubit. Figure 2(c) shows the experimental spectra obtained by reading out the C qubit. The spectrum consists of four resonance peaks, labeled by the corresponding logical states of the corresponding logical states and of nuclei H and F. With respect to the status of probe qubit H, we divide the four signal peaks into two groups: Group 1, including peaks 1⃝ and 3⃝, corresponds to state of H, indicating the failure cloning process; and group 2, including peaks 2⃝ and 4⃝, corresponds to state , indicating the success process. Unlike approximate quantum cloning, the probabilistic cloning machine will yield perfect cloning, and the faulty copies are rejected. In the analysis, we filtered data from group 1, which represents the failing case in the cloning process.
The signal intensity of the initial pseudopure state is measured as the normalized reference. Then the relative signal intensity is measured by digital quadrature detection (DQD). The observable operator can be expressed as , so the transverse components can be measured as and . In the subsequent experiment, the vertical part is measured by applying a pulse on the qubit after the cloning process.
It is well-established that probabilistic cloning machine are analyzed with respect to two characteristics: cloning efficiency and cloning fidelity . In the following, we will discuss in details how we can obtain these two parameters from the spectrum data. In the experiment, cloning efficiency equals to the population of the probe qubit H in the state . It can be measured by comparing the signal intensity of group 2 with the total one, as shown in Eq.(NMR Experimental Demonstration of Probabilistic Quantum Cloning).
We have studied the input states as the function of , varied from to in increment. Fig.4 display the average cloning efficiency (green columns) along with the theoretical expectations (red columns). Note that the probabilistic quantum cloning machine can produce faithful copies with probability . The cloning efficiency is related to the distinguishable metric of the quantum state space for the input states, since it increases with decreasing of the overlap of the input states in the cloning set . The result clearly indicates that the larger the overlap between the input states, the smaller the maximum cloning efficiency. As the angle approaches , the cloning efficiency will be close to 1, which is possible if and only if the states are chosen from an orthogonal set.
After the experimental cloning efficiencies were determined, another important parameters, cloning fidelities, were analyzed. From the Bloch-sphere representation, the state of a single qubit can be represented by a density matrix of the form , where is the identity operator and are usual Pauli matrices. Let be the density matrix for the initial state, while for the copies. To experimentally determine the fidelities, we need the density operators of the final states of both qubits. The length of the vectors on the Block sphere representing the cloned states are found to be , where (). Here, the signal intensity should be divided by to compensate the signal loss of faulty copies. The cloning fidelities are expressed as
For the initial state , we have and the fidelities become
The experimental fidelities of the duplicated states are shown in Fig.5. Fig.5(b)-(d) show the reconstructed density matrices for . The vertical axes show the normalized amplitude and the horizontal axes label the basis state in the computational basis. Fig.5(b) represents the matrix for the input state and (c), (d) give the experimental results of and . The corresponding fidelities are , . Fig.5(a) shows the cloning fidelities for the input states with different angles in the cloning set . In the figure, the cycles denote the cloning fidelities for the second qubit (red) and third qubits (green) for , while the triangles are for . The average experimental fidelity over different values of is about . The small deviations () between the experimental and theoretical values are mainly attributed to imperfect calibration of radio frequency pulses. The decoherence from spin relaxation is negligible, since the total experimental time of ms is much shorter than the minimal relaxation time of s.
In summary, we have experimentally demonstrated the probabilistic cloning machine in a NMR system by simplifying the network. For two orthogonal states, the cloning machine can always produce two perfect copies with the 100% probability of success. When the original states are non-orthogonal, two faithful copies can be produced with the deterministic probability less than 1. The experimental results are in good agreement with the the theoretical prediction, which indicates the cloning network is effective for the all input states. Our experimental scheme can be achieved not only in NMR systems, but also in other physical systems. Further research works in probabilistic cloning, especially in experimental demonstration, will be important for the other quantum information protocols, such as quantum identification, quantum purification and quantum deleting Duan80 (); Fiuravsek70 (); Feng65 ().
We thank Prof. J. W. Pan and Dr. W. Harneit for helpful discussions. This work was supported by the National Natural Science Foundation of China, the National Science Foundation for Postdoctoral Scientists, the Fundamental Research Funds for the Central Universities, the CAS, Ministry of Education of PRC, and the National Fundamental Research Program.
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