Abstract
Quantum mechanics provides a disembodied way to transfer quantum information from one quantum object to another. In theory, this quantum information transfer can occur between quantum objects of any dimension, yet the reported experiments of quantum information transfer to date have mainly focused on the cases where the quantum objects have the same dimension. Here, we theoretically propose and experimentally demonstrate a scheme for quantum information transfer between quantum objects of different dimensions. By using an optical qubit-ququart entangling gate, we observe the transfer of quantum information between two photons with different dimensions, including the flow of quantum information from a four-dimensional photon to a two-dimensional photon and vice versa. The fidelities of the quantum information transfer range from 0.700 to 0.917, all above the classical limit of 2/3. Our work sheds light on a new direction for quantum information transfer and demonstrates our ability to implement entangling operations beyond two-level quantum systems.
© 2022 Chinese Laser Press
1. INTRODUCTION
The information transfer between different objects is one of the most fundamental phenomena in nature. In the classical world, a macroscopic object that carries unknown information can have its information precisely measured and copied, and thus this information can be transferred to another object while still being retained on the original object. In the quantum world, although quantum mechanics does not allow the unknown quantum information carried on a quantum object to be perfectly cloned or precisely measured [1,2], it does allow quantum information to be transferred from one object to another object in a disembodied way, i.e., only the quantum information but not the object itself is transferred. Quantum information transfer (QIT) between two quantum objects, which is also called quantum teleportation [3,4] when the two objects are separated at different locations, is widely used in quantum information applications including long-distance quantum communication [5–7], distributed quantum networks [8,9], and measurement-based quantum computation [10–14]. It has been experimentally demonstrated in a variety of physical systems [15–27], including photons [17,18], atoms [19], ions [20–22], electrons [23], defects in solid states [24], optomechanical systems [25], and superconducting circuits [26,27]. Recently, more complex experiments have also been reported, such as the open destination teleportation [28,29] and the teleportation of a composite system [30,32], a multilevel state [31,32], and multidegree of freedom of a particle [33].
So far, the reported experiments have mainly focused on QIT between quantum objects with the same dimension. However, in quantum applications such as distributed quantum networks, different quantum objects may have different dimensions, and QIT between them is also required. For example, as shown in Fig. 1(a), there are two quantum objects and , where is two-dimensional (2D) and carries no quantum information, and is four-dimensional (4D) and carries two qubits of unknown quantum information beforehand. If wants to transfer quantum information to , because is 2D and capable of carrying at most one qubit of quantum information, only one of the two qubits stored in can be transferred to . We call this process a 4-to-2 QIT, which will distribute the two qubits of quantum information originally concentrated on over both and . Now and are each loaded with one qubit of quantum information. Obviously, as shown in Fig. 1(b), this one qubit of quantum information stored in can also be transferred back to . We call this process a 2-to-4 QIT, which concentrates the two qubits of quantum information distributed over both and on object only.
In this work, we theoretically propose and experimentally demonstrate a scheme for QIT between a 2D quantum object and a 4D one. By using an optical qubit-ququart entangling gate, we successfully transfer one qubit of quantum information from a 4D photon preloaded with two qubits of quantum information to a 2D photon, i.e., achieving a 4-to-2 QIT. We also experimentally realize a 2-to-4 QIT, i.e., transferring one qubit of quantum information from a 2D photon to a 4D photon preloaded with one qubit of quantum information. Besides fundamental interests, the QITs demonstrated here have the potential to simplify the construction of quantum circuits and find applications in quantum computation and quantum communications.
2. SCHEME OF QUANTUM INFORMATION TRANSFER
A. 2-to-2 QIT
Before presenting the scheme for implementing the QIT between quantum objects of different dimensions, let us briefly review how to implement the QIT between quantum objects of the same dimensions. Here we take the 2D case as an example, assuming that both and are 2D quantum objects, where is in the state without any quantum information preloaded, and is in the state loaded with one-qubit unknown quantum information. As shown in Fig. 2(a), by applying a controlled- gate ( gate, commonly referred as CNOT gate) on and , the quantum state of the composite system of and would become
where , . After measuring in the basis and forwarding the measurement outcome to , a unitary operation ( or ) based on the outcome is applied on and thus converts its state to . The state of now has the same form as the initial state of and thus completes the QIT from to .B. 4-to-2 QIT
We now present the scheme for realizing the QIT from a 4D to a 2D quantum object. Suppose is still 2D and in the state , and is now 4D and in the state
which is preloaded with two-qubit unknown quantum information. To achieve the QIT between a 2D and a 4D quantum object, instead of using a two-qubit gate like the CNOT gate, one should use a qubit-ququart entangling gate. As shown in Fig. 2(b), a controlled- gate () is applied to and , where is a 4D unitary gate defined as which converts () to () and vice versa. The state of and is thus converted to A projective measurement is then applied on to measure whether it is in the subspace spanned by and or the subspace spanned by and . Based on the measurement outcome, unitary operations ( or ) are applied on and and their state becomes where Comparing Eq. (2) with Eq. (1), it is observed that the two quantum states have exactly the same form except for the difference in the state basis, which means that the two-qubit quantum information previously stored in is now distributed over both and . In other words, one of the two qubits of quantum information originally stored in is now transferred to , thus achieving a 4-to-2 QIT.C. 2-to-4 QIT
We now show how the same quantum circuit can be used to implement a 2-to-4 QIT (the inverse of the above process), i.e., transferring one qubit of quantum information from a 2D quantum object to a 4D quantum object preloaded with one-qubit unknown quantum information. The initial state of and can be written as
where both the 2D and the 4D are preloaded with one qubit of unknown quantum information. Note that the quantum state of and can be either an entangled state or a separable state. As shown in Fig. 2(c), a gate is applied to and , and their state is thus converted to where . Similarly, after measuring in the basis and forwarding the outcome to , a 4D unitary operation ( or ) is applied on conditioned on the outcome and thus converts its state to where Comparing Eq. (4) with Eq. (3), it is observed that the two quantum states have exactly the same form except for the difference in the state basis, which means that the two qubits of quantum information previously distributed over both and are now concentrated in . In other words, the one-qubit quantum information originally stored in is now transferred to , thus achieving a 2-to-4 QIT.3. EXPERIMENTAL DEMONSTRATION USING LINEAR OPTICS
A. Optical Gate
To experimentally implement the QIT operations described above, the main challenge lies in the realization of the key part of the quantum circuit, namely the qubit-ququart entangling gate . Most experimentally realized quantum entangling gates so far are based on qubits [34–40], and it is a challenging task to implement such high-dimensional entangling operations in any physical system.
Here we present our method of implementing the gate using linear optics. As shown in Fig. 3(a), instead of implementing the gate directly, we first decompose it into two consecutive gates and based on the fact that , where
Although () is a 4D unitary operation, it only operates on a 2D subspace spanned by and ( and ). () swaps and ( and ) and leaves and ( and ) unchanged. Based on this fact, two optical CNOT gates can be used to implement and .As shown in Fig. 3(a), system consists of two photons, and , serving as the control qubit, and system consists of photon , serving as the target ququart. The two orthonormal basis states of system are and , where and denote horizontal and vertical polarizations, respectively. For system , to encode a ququart with a single photon, photon , both the polarization and spatial degrees of freedom are used, and the four orthonormal basis states are , , , and , where () denotes photon in the upper (lower) spatial mode with horizontal polarization, and () denotes photon in the upper (lower) spatial mode with vertical polarization. The operation between qubit and ququart can be realized by applying a polarization CNOT gate to photon and photon in the upper path, which can be understood as follows. When the polarization of photon is (namely qubit in ), nothing happens; when the polarization of photon a is (namely qubit in ), the polarization of photon flips between and if is in the upper path (namely ququart ’s and components being swapped), and is unaffected if is in the lower path (namely ququart ’s and components being unchanged), which is exactly what a gate achieves. Similarly, the operation can be realized by applying a polarization CNOT gate to photon and photon in the lower path. As a result, the desired gate can be achieved by the two polarization CNOT gates as shown in Fig. 3(a).
Note that the use of two photons to encode the control qubit is mainly due to the following experimental considerations. The optical CNOT gates used experimentally are based on postselection measurements, and such CNOT gates would fail when they act on the same two photons twice in a row. By using two photons and to encode the control qubit, the two CNOT gates are not acting on the same two photons, thus avoiding this problem.
B. Experimental Setup
Figure 3(b) shows the experimental setup for implementing the two QIT schemes. Two photon pairs are generated by passing femtosecond-pulse UV laser through type-II beta-barium borate (BBO) crystals (see Appendix A). Photons and of the first pair are prepared at , which serves as the initial state of system . By passing through the beam displacer BD1 and its surrounding waveplates (HWP2, QWP2, HWP3, QWP3, HWP4, and QWP4), photon from the second pair is prepared at , which serves as the initial state of system . The upper (lower) rail of photon is then superposed with photon () on a partial polarization beamsplitter (PPBS). The PPBS, the loss elements, and its surrounding half wave-plates can realize a polarization CNOT gate on photon () and photon in the upper (lower) path [41]. These two polarization CNOT gates together realize a gate on qubit and ququart (see Appendix C), and the state of the three photons becomes
C. Results of the 4-to-2 QIT
To demonstrate the 4-to-2 QIT, and are set to , and the three-photon state after the gate can be written as
Active feed-forward is needed for a full, deterministic 4-to-2 QIT. However, in this proof-of-principle experiment, we did not apply feed-forward but used postselection to realize a probabilistic 4-to-2 QIT. By postselecting the and components and converting to and to using BD2 and its preceding waveplates, the three-photon state is obtained. By projecting photon to , the two-photon state of and becomes The two qubits of quantum information originally concentrated on photon are now distributed over two photons and , which indicates that one qubit of quantum information has been transferred from photon to photon , thus achieving a 4-to-2 QIT.We then measure the fidelity of the final state, , which is defined as the overlap between the ideal final state () and the measured density matrix (). The verification of the QIT results is based on fourfold coincidence detection which in our experiment occurs with a rate of 0.22 Hz. In each setting, the typical data collection time is 10 min, which allows us to sufficiently suppress Poisson noise.
Five different initial states of are prepared for demonstrating the 4-to-2 QIT:
Figures 4(a)–4(e) shows the 4-to-2 QIT results of the five different initial states on specific bases from which the fidelities can be extracted. For each of the five initial ququart states to , the fidelity of the final state of and is, in numerical sequence: , , , , and , which are summarized in Fig. 4(f). The fluctuation of the fidelities in the experiment stems from the fact that the realized CNOT gates have different performance for different control qubits, i.e., the noise is higher when the polarization of the control qubit is horizontal.D. Results of the 2-to-4 QIT
To demonstrate the 2-to-4 QIT, and are set to zero, and the quantum state after the gate can be written as
By projecting both photons and to , the quantum state of photon becomes This ququart state of photon is then analyzed by the measurement setup consisting of BD2, its surrounding waveplates, the polarization beamsplitter (PBS), and the single-photon detector D3 (see Appendix D). The two qubits of quantum information originally distributed over system (photons and ) and system (photon ) are now concentrated on system , which indicates that one qubit of quantum information has been transferred from system to system , thus achieving a 2-to-4 QIT.Nine different initial states of and are used in the general quantum state transfer experiment:
Figures 5(a)–5(i) show the 2-to-4 QIT results of the nine different initial states on specific bases, from which the fidelities can be directly extracted. For each of the nine initial states to , the fidelity of the final state of ququart is, in numerical sequence: , , , , , , , , and , which is summarized in Fig. 5(j).The reported data are raw data without any background subtraction, and the main errors are due to double pair emission, imperfection in preparation of the initial states, and the nonideal interference at the PPBS and BD2. Throughout our experiments, whether 2-to-4 or 4-to-2, we are actually transmitting one qubit of quantum information. Specifically, in the 4-to-2 experiment, particle is preloaded with two qubits of quantum information and sends one of two qubits to . In the 2-to-4 experiment, the one qubit carried by is transmitted to which is already preloaded with one qubit of quantum information. Therefore, the classical threshold should be in our experiments. Despite the experimental noise, the measured fidelities of the quantum states are all well above the classical limit , defined as the optimal state-estimation fidelity on a single copy of a one-qubit system [42]. These results prove the successful realization of the 4-to-2 and the 2-to-4 QIT.
4. CONCLUSION
In this work, we have experimentally transferred one qubit of quantum information from a 4D photon preloaded with two qubits of quantum information to a 2D photon. We have also experimentally realized the inverse operation, namely transferring one qubit of quantum information from a 2D photon to a 4D photon preloaded with one qubit of quantum information. Our experiments show that quantum information is independent of its carriers and can be freely transferred between quantum objects of different dimensions. Although the present experiments are realized in the linear optical architecture, our protocols themselves are not limited to the optical system and can be applied to other quantum systems such as trapped atoms [19], ions [21,22], and electrons [23].
The techniques developed in this work for entangling operations on photons of different dimensions can be used to prepare a new type of maximally entangled state such as , where photons and are both 2D and belong to system , and photon is 4D and belongs to system . System and system have the same dimension but a different number of particles. Such maximally entangled states with asymmetric particle numbers can be used as a physical resource for realizing quantum teleportation between systems with the same dimension but different particle numbers.
Our approach can be readily extended to higher dimensional cases (see Appendixes E and F). With these QIT operations, one can either concentrate the quantum information from multiple objects to one object or distribute the quantum information from one object to multiple objects. Such operations have the potential to simplify the construction of multiqubit gates [43,44] (see Appendix G) and find applications in quantum computation and quantum simulations.
APPENDIX A: GENERATING TWO PHOTON PAIRS
For the sake of simplicity, in Fig. 3(b) of the main text, we only show a simplified version of the spontaneous parametric down-conversion (SPDC) sources. Figure 6 shows the detailed experimental setup for generating two photon pairs. An ultraviolet pulse laser centered at 390 nm is split into two parts, which are used to generate two SPDC photon pairs. The lower part of the laser directly pumps a BBO crystal to generate a pair of photons in the state via beamlike type-II SPDC, where photon is used for the trigger. The upper part of the laser goes through HWP1 and QWP1 to prepare its polarization at . It then passes through an arrangement of two beam displacers (BDs) and HWPs to separate the laser into two beams by 4 mm apart (such configuration was first adopted by Zhong et al. in Ref. [45]). The two beams then focus on a BBO crystal to generate two photon pairs in the states and via beamlike type-II SPDC, where the subscripts denote the spatial modes. and are then rotated using HWPs to and , respectively. Photon pairs of and are then combined into the same spatial modes using two BDs. After tilting the two BDs to finely tune the relative phase between the two components, the two photons and are prepared into , which is the desired quantum state of system .
APPENDIX B: TWO-PHOTON INTERFERENCE ON A PPBS
The PPBS implements the quantum phase gate by reflecting vertically polarized light perfectly and reflecting (transmitting) 1/3 (2/3) of horizontally polarized light. To realize a perfect quantum gate with the PPBS, the input photons on the PPBS need to be indistinguishable to each other. To evaluate the indistinguishability of the input photons, a two-photon Hong-Ou-Mandel (HOM) interference on the PPBS needs to be measured. For large delay, the two photons are completely distinguishable due to their time of arrival. The probability to get a coincidence from an input is then 5/9. In case of perfectly indistinguishable photons at zero delay, the probability drops to 1/9. From the above considerations, the theoretical dip visibility is obtained, which is defined via , where is the count rate at zero delay, and is the count rate for large delay. As shown in Fig. 7, the HOM interference is experimentally measured, and a dip visibility of is obtained, where the error bar is calculated from the Poissonian counting statistics of the detection events. The overlap quality indicates that about of the detected photon pairs are distinguishable.
APPENDIX C: IMPLEMENTATION OF GATE USING LINEAR OPTICS
As described in the main text, an optical gate between system (photons and ) and system (photon ) can be implemented with a setup as shown in Fig. 8(a). Photons and encode the control qubit, and its initial state is . Photon encodes the target ququart, and its initial state is . After passing through the loss elements that transmit horizontally polarized light perfectly and transmit 1/3 of vertically polarized light, photon () and photon in the upper (lower) mode are superposed on the PPBS. The PPBS, the loss elements, and the two surrounding HWPs at 22.5° together implement a polarization CNOT operation on the input photons [41,46,47]. Such optical circuit can thus realize the following transformations:
As a result, after passing through this optical circuit, the initial input state would become which is exactly the desired output state of a gate. The above optical gate operates with a success probability of 1/27. In practice, to combat low count rates, we adopt the method proposed in Ref. [41] to simplify the implementation of the optical gate. The simplified experimental setup is shown in Fig. 8(b). We achieve a correct balance by removing the loss elements and prebiasing the input polarization states during gate characterization. The initial state of photons and is now prepared at instead of . The HWP applied on photon before entering the PPBS is now set at 15°, thus converting photon to the state instead of , which is the case if the HWP is set at 22.5°.APPENDIX D: STATE ANALYSIS OF A PHOTONIC QUQUART STATE
A photon with both polarization and spatial degrees of freedom (DOFs) can encode a ququart state. To fully characterize such state, one needs to perform projective measurement onto various different ququart states. To fulfill this task, we build a ququart state analyzer as shown in Fig. 9, which can project the input ququart to any state in the form of . This setup works as follows. Suppose the input ququart state is . After passing through QWP3 and HWP3, which are used to convert to , the ququart state becomes . The subsequent two HWPs (one at 45° and the other at 0°) and the BD are used to convert to , which is now a polarization qubit state. QWP4 and HWP4 are then used to convert to , which can pass through the PBS and get detected by the SPD. As a result, for any input ququart state, only its component can pass through the setup described above, which effectively realizes the desired projective measurement. By changing the parameters , , , and , this state analyzer can be used to perform a full state tomography on the input ququart state.
APPENDIX E: MERGE OPERATION
The scheme of 2-to-4 QIT can be extended to the higher dimensional case, i.e., transferring one qubit of unknown quantum information from a qubit to a qudit. We call the operation of aggregating quantum information from two particles to one particle the Merge operation, and Merge() denotes the aggregation of quantum information of a qubit and a -dimensional qudit to a -dimensional qudit. The quantum circuit to implement Merge() is shown in Fig. 10(a), where the initial state of system is , and the initial state of system is , which is a -dimensional qudit. A controlled- () gate is applied to and , where is a -dimensional unitary gate defined as , which swaps and for in the range of 0 to , expanding the state space of system from to dimensions. The state of and is thus converted from
to After measuring in the basis and forwarding the outcome to , a -dimensional unitary operation ( or ) is applied on conditioned on the outcome and thus converts its state to where By comparing Eq. (E2) and Eq. (E1), one sees that the two quantum states have exactly the same form except for the difference in the state basis, which means that the quantum information originally stored in and has been merged into .APPENDIX F: SPLIT OPERATION
The scheme of 4-to-2 QIT can be extended to the higher dimensional case, i.e., transferring one qubit of unknown quantum information from a qudit to a qubit. We call this operation of distributing quantum information from one particle to two particles the Split operation, and Split() denotes the distribution of quantum information from a -dimensional qudit to a qubit and a -dimensional qudit. The quantum circuit to implement Split() is shown in Fig. 10(b), where system is initially in the state , and the initial state of system is in the -dimensional qudit state:
A gate is applied to and , and their state becomes A projective measurement is then applied on to measure whether it is in the subspace spanned by , , and or the subspace spanned by , , and . Based on the measurement outcome, unitary operations ( or ) are applied on and , and their state becomes Comparing Eq. (F2) and Eq. (F1), it is observed that the two quantum states have exactly the same form except for the difference in the state basis, which means that the quantum information originally stored in is now split into and .APPENDIX G: CONSTRUCTION OF MULTIQUBIT GATES
In various quantum information applications, including quantum computation and quantum simulation, multiqubit quantum gates are widely used. Theoretically, multiqubit quantum gates can be decomposed into two-qubit CNOT gates and single-qubit quantum gates for implementation, but in practice, such decomposition can be quite complex and consumes a lot of resources experimentally. Here, we propose a method to simplify the implementation of multiqubit quantum gates by using QIT methods. This approach essentially transforms an arbitrary -qubit quantum gate operation into a unitary transform on a -dimensional qudit, which is simpler to implement in certain circumstances. For example, for a path-encoded -dimensional photon, when is not very large, an arbitrary qudit unitary transform can be readily implemented using the Reck scheme [48,49]. We present below our method using a 3-qubit quantum gate as an example. Specifically, the method can be divided into three steps. (1) The quantum information of the three input qubits is converged to a single particle, which can be achieved by two Merge operations. As shown in Fig. 10(c), a ququart is obtained by a Merge(2, ) operation acting on qubit 2 and qubit 1. Then a Merge(2, ) operation acts on qubit 3 and this ququart to obtain a qudit (), which contains the quantum information of the input three qubits. (2) The qudit is then subjected to an unitary transformation, which has the same mathematical form as the matrix of the three-qubit quantum gate expanded in the computational basis. (3) The quantum information of this qudit is then distributed to three particles, which can be achieved by two Split operations. The eight-dimensional qudit is split by a Split() operation to get a qubit and a ququart, and this ququart is then split by a Split() operation to finally get three qubits at the output, thus completing the three-qubit quantum gate. For an arbitrary -qubit quantum gate, it is often simpler and more resource-efficient to use our method than the traditional decomposition of -qubit quantum gates into CNOT gates and single-qubit quantum gates, as long as is not very large, i.e., the -dimensional qudit can be easily unitary-transformed.
Funding
National Natural Science Foundation of China (61974168); Special Project for Research and Development in Key Areas of Guangdong Province (2018B030329001, 2018B030325001); National Key Research and Development Program of China (2017YFA0305200, 2016YFA0301300).
Disclosures
The authors declare that there are no competing interests.
Data Availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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