Weaving an entangled cluster
Entanglement is a powerful resource for quantum computation and information processing. One requirement is the ability to entangle multiple particles reliably. Schwartz et al. created an on-demand entangled cluster state of several photons by addressing a quantum dot with a sequence of laser pulses (see the Perspective by Briegel). They used an internal state of the quantum dot, a dark exciton, and its association with another internal state, a biexciton, to weave successive photons into an entangled cluster, generating entanglement between up to five photons.
Abstract
Photonic cluster states are a resource for quantum computation based solely on single-photon measurements. We use semiconductor quantum dots to deterministically generate long strings of polarization-entangled photons in a cluster state by periodic timed excitation of a precessing matter qubit. In each period, an entangled photon is added to the cluster state formed by the matter qubit and the previously emitted photons. In our prototype device, the qubit is the confined dark exciton, and it produces strings of hundreds of photons in which the entanglement persists over five sequential photons. The measured process map characterizing the device has a fidelity of 0.81 with that of an ideal device. Further feasible improvements of this device may reduce the resources needed for optical quantum information processing.
The concept of entanglement is a fundamental property of quantum mechanics (1) and an essential ingredient of proposals in the emerging technologies of quantum information processing (2, 3), including quantum communication (4, 5) and computation (6, 7). In many cases, these applications require multipartite entanglement, encompassing a large number of quantum bits (qubits). Such entanglement is often very fragile and can be adversely affected, or even completely vanish, when one of the qubits interacts with its environment or is lost from the system. A special class of quantum states exhibits a persistence of their multipartite entanglement (8). The entanglement for this class of states is robust to adverse effects on a subset of the qubits. A prominent example of such a quantum state is a cluster state (8)—a string of mutually entangled qubits. Cluster states serve as an important resource for quantum computing, allowing its implementation solely via single-qubit measurements (9). A photonic implementation of cluster states enjoys many advantages, due to the noninteracting nature of photons that suppresses decoherence effects, as well as the precise single-qubit measurements provided by linear optics technology. Generating cluster states is, however, a formidable task. Previously, cluster states of finite size have been demonstrated in trapped ions (10, 11) and in continuous-variable modes of squeezed light (12). In addition, photonic cluster states have been obtained using frequency down-conversion techniques (13–16). Despite these demonstrations, the quest for obtaining a scalable, deterministic source of cluster states is still under way. Here, we use quantum dots (QDs), “artificial atoms,” which are on-demand sources of both single photons (17, 18) and entangled photon pairs (19–21), to implement a scheme for deterministic generation of long strings of entangled photons in a cluster state (22). We demonstrate a prototype device that produces strings of a few hundred photons in which the entanglement persists over five sequential photons. Feasible improvements of the device provide a route for both determinism and scalability. Our implementation thereby forms a building block for future quantum information processing developments such as measurement-based quantum computation and quantum communication.
Experimentally realizing a cluster state
Our implementation is based on a proposal of Lindner and Rudolph (22) in which repeated timed optical excitations of a confined electron in a single semiconductor QD result in the formation of a cluster state composed of the sequentially emitted single photons. The proposal uses the spin of the electron as a matter spin qubit, whose state is entangled with the polarization of the emitted photon (19, 23–26) resulting from the optical excitation. This excitation can, in principle, be repeated indefinitely, whereas the precessing electron spin acts as an “entangler” and entangles the emitted photons to produce a one-dimensional (1D) cluster state.
We present a practical realization of this proposal, in which the QD-confined electron is replaced by a confined dark exciton (DE) (27–29). The DE is a semiconductor two-level system, effectively forming a matter spin qubit [see supplementary materials (SM) section 3 (30)]. The two 
DE spin projections on the QD symmetry axis 
form a basis, 
, for the DE space (Fig. 1, A and B). The DE energy eigenstates are 
, with an energy splitting 
corresponding to a precession period of 
nsec (27, 28). In addition to the DE, our experiment uses two states of a biexciton (BiE)—a bound state of two excitons—whose total spin projections on 
are either 
or 
, with a precession period of 
nsec. We denote these states by 
. The experimental protocol relies on the optical transition rules 
and 
through right 
and left 
hand circularly polarized photons, respectively (27, 28), in direct analogy with the original proposal (22). The energy-level diagram describing the DE, the BiE, and the optical transition rules is schematically summarized in Fig. 1C.
(A) The QD containing a DE in a microcavity. The optical axis (indicated by a blue arrow for incoming laser light and a magenta arrow for outgoing emitted light) is parallel to the QD symmetry axis. (B) Bloch sphere representation of the DE spin and photon polarization qubits. Red and blue define the bases for the DE and photon states, respectively. (C) The DE states (
), BiE states (
), and optical transition rules between the states. Upward arrows represent resonant excitation. Downward arrows represent photon emission. The gray circular arrows represent precession of the DE and BiE states. (D) The sequence of transitions required to generate a cluster state. The green arrow represents the initialization pulse that generates a DE in spin eigenstate. Repeated timed excitations of the DE to the BiE (blue arrows) result in repeated emission of single photons (magenta arrows). Correct interpulse spacing results in an entangled cluster state. The lower panel is a circuit representation of the resulting 1D cluster state (22). Each horizontal line represents a qubit. The uppermost line represents the DE, and the lines below represent timed ordered emitted photons. 
and 
represent DE spin and photon polarization states. DE initialization is represented by the gate 
, and photons are initialized in the fiducial state 
. The timed DE precession is represented by the single-qubit gate 
, and excitation-emission is represented by a 
gate (vertical line) between the DE and the emitted photon. The area enclosed in the dashed box represents one unitary cycle in the ideal protocol.
An ideal protocol for cluster-state generation
Before the protocol begins, the DE is deterministically initialized in its higher-energy spin eigenstate 
(31) by a short 
-area picosecond pulse (29). A 
-area pulse transfers the entire population from one quantum state to another.
The protocol, which begins immediately after the initialization, consists of repeated applications of a cycle. The cycle contains three elements: (i) a converting laser 
-pulse, resonantly tuned to the DE-BiE optical transition. The pulse is rectilinearly horizontally 
polarized and coherently converts the DE population to a BiE population; (ii) subsequent radiative recombination of this BiE, resulting in a DE in the QD and emission of a photon; and (iii) timed free precession of the DE spin. This cycle can be applied multiple times to generate an entangled multiphoton cluster state. Figure 1D illustrates the above procedure and its equivalent circuit diagram. One full cycle of the protocol is indicated in the circuit diagram by a dashed rectangle.
The state generated by the above protocol is revealed by following the evolution of the system during the first three steps after the initialization. In step (i), the horizontally polarized 
-area pulse “converts” the DE coherent state into a coherent BiE state: 
. In step (ii), radiative recombination of this BiE results in an entangled state of the emitted photon polarization and the DE spin (28), 
, where 
is the first photon right- (left-) hand circular polarization state. The excitation and subsequent photon emission are represented in Fig. 1D by a 
gate between the DE and the emitted photon. In step (iii), the DE precesses for 
of a precession period. The precession is represented in Fig. 1D by the single-qubit gate 
, where 
is the corresponding Pauli matrix. The sequential application of the 
and 
gates forms one full cycle in our protocol. In the beginning of the next cycle, the DE-photon state is given by
(1)The above cycle is now repeated: reexcitation (27, 28) to the BiE state, recombination of the second BiE, and timed precession associated with the 
gate. This results in a second photon, whose polarization state is entangled with that of the first photon and the spin of the remaining DE, yielding the tripartite state
(2)Repetition of the reexcitation-emission and subsequent precession cycle generates a 1D string of polarization-entangled photons in a cluster state, as shown in the equivalent circuit diagram of Fig. 1D. We have realized the above protocol in which the cycles were implemented with fidelity of 0.81 to the ideal cycle described above.
Considerations in practical realizations of the protocol using the dark exciton as entangler
The DE has many advantages as an entangler for sequential generation of entangled photons. It exhibits a long lifetime (
nsec) and a long coherence time (
nsec) (28). In addition, the DE spin state can be deterministically written in a coherent state using one single short optical pulse (28, 29) and can be reset (i.e., emptied from the QD) using fast all-optical means (32). Furthermore, the DE to BiE excitation resonance occurs at a higher energy than the BiE to DE main emission resonance (SM, section 3), thereby facilitating accurate background-free single-photon detection. In addition, the generated cluster state is unaffected by the coherence of the photons’ wave packets (22).
Despite these advantages, several types of imperfections must be considered. The dominant imperfection originates from the finite BiE radiative lifetime, 
nsec (28). Because the DE and BiE precess during the emission process, the purity of the polarization state of the photons is reduced (22). Another type of imperfection is the decoherence of the DE spin during its precession, resulting from the hyperfine interaction between the DE and nuclear spins in the semiconductor (22). Therefore, to ensure generation of a high-quality cluster state, three important parameters should be kept small: the ratio between the BiE radiative time 
and the DE and BiE precession times 
and 
, and the ratio between the DE precession time and its decoherence time 
. In our system, 
and 
. Because all these parameters are much less than unity, the implemented protocol has high fidelity to the ideal one, as we now show.
Demonstrating an entangled cluster state
The demonstration that our device generates an entangled multiphoton cluster state is done in two complementary steps. First, we determine the nonunitary process map acting in each cycle of the protocol, which replaces the 
and 
unitary gates of Fig. 1D. The process map is a linear map from the initial DE qubit’s space to the space of two qubits comprising the DE and the newly emitted photon. It fully characterizes the evolution of the system in each cycle of the protocol, thereby completely determining the multiphoton state after any given number of cycles. Then, we verify that the three-qubit state, consisting of the DE and two sequentially emitted photons, generated by applying two cycles of our protocol is a genuine three-qubit entangled state. We also quantify the degree of entanglement between each of the three pairs of qubits.
To measure the process map, we perform quantum process tomography. We first initialize the DE in four different states, 
, 
, 
, and 
. The states are defined in Fig. 1B. In reality, the initialization is in a partially mixed state (SM, section 4). For each DE initialization, we apply one cycle of the protocol and perform correlation measurements between the resulting emitted photon polarization and the DE spin. In these correlation measurements, we project the emitted photon on the polarization states 
, 
, 
, and 
, while making projective measurements of the DE’s spin state. For the DE projective measurements, we apply either a right- or a left-hand circularly polarized 
-area pulse at the end of the cycle. Due to the optical selection rules (Fig. 1C), this pulse deterministically excites either the 
or the 
DE to the BiE, respectively. Detection of an emitted photon after this excitation projects the DE spin on the states 
or 
at the time of the pulse. To project the DE onto the spin states 
or 
at the end of the cycle, we rotate the DE state by delaying the pulse a quarter of a precession period. This method allows us to project the DE on the states 
and 
but not on the 
states. Therefore, by the above two-photon correlation measurements, we directly measure only 48 out of the required 64 process map matrix elements.
To complete the quantum process tomography, we use three-photon correlation measurements. For each of the four initializations of the DE, we apply two cycles of the protocol. We then perform full polarization state tomography between the resulted two emitted photons, while projecting the spin of the remaining DE on 
and 
, as discussed above. The two- and three-photon correlations, which together uniquely determine all the 64 matrix elements of the process map, are fully described in sections 7 and 8 of the SM.
In Fig. 2A, we present the measured process map, and in Fig. 2B, we present the process map corresponding to the ideal protocol. The fidelity (see SM, section 10, for definition) between the measured and ideal process map is 
. Figure 2C gives the process map obtained by modeling the evolution of the system using a master equation approach with independently measured system parameters (SM, section 5). The fidelity between the model and the measured process map is 
. These fidelities indicate that our device is capable of deterministically generating photonic cluster states of high quality, thus providing a new resource for quantum information processing.
The process map 
, which describes the evolution of the system in each cycle of the protocol, is a completely positive and trace-preserving map from the DE one-qubit space to the two-qubit space composed of the DE and the emitted photon. The process map acts trivially on all other photons in the state. We use the convention 
, where 
is the density matrix that describes the DE state. The sums are taken over 
, where 
is the identity matrix and 
are the corresponding Pauli matrices. The 64 real parameters 
thus fully specify 
. (A) The process map as measured by quantum process tomography. The values of 
, are presented such that the rows correspond to the indices 
and the columns correspond to the index 
. (B) The unitary process map corresponding to the ideal protocol as in Fig. 1D. (C) The process map as calculated from a theoretical model with independently measured parameters (SM, section 5).
Using the measured process map, we can now describe the state 
, consisting of the DE and N photons, following successive applications of N protocol cycles. An important method to quantify the multipartite entanglement in 
is to consider the maximal degree of entanglement between two chosen qubits once the rest of the qubits are measured. The resulting quantity is referred to as the localizable entanglement (LE) between the two chosen qubits (33). In an ideal cluster state, the localizable entanglement is maximal: Any two qubits can be projected into a maximally entangled state by measuring the rest of the qubits. For example, consider the three-qubit state described by Eq. 2. By projecting the DE on the 
state, we obtain two photons in the maximally entangled state
(3)To compute the LE between two qubits in the state 
, we obtain their reduced density matrix after the rest of the qubits have been measured in an optimized basis. The degree of entanglement between the two qubits is then evaluated by a standard measure: the negativity (34) 
of their reduced density matrix. 
is defined as the magnitude of the negative eigenvalue of the partially transposed density matrix. For 
the qubits are entangled, and 
corresponds to maximally entangled qubits. Full definitions for the negativity and LE are given in the SM, section 6.
In Fig. 3, we plot using blue circles the LE in the state 
, obtained from the measured process map, as a function of the distance between two qubits in the string. As expected, the LE in the 1D state 
decays exponentially with the distance between the chosen two qubits (33). The LE, which characterizes the robustness of the multipartite entanglement in the state produced by our device, is shown to persist up to five qubits.
The horizontal axis indicates the distance d between two qubits in the string (the qubits are ordered according to the labeling convention in Fig. 1D), and the vertical axis corresponds to the negativity of the reduced density matrix of the two qubits. The blue circles correspond to the LE between qubits m and m + d (
) in the state 
obtained using the measured process map (SM, section 6). Here, we use 
and 
, but other values for 
and 
yield similar results. The dashed lines represent the best fit to an exponential decay law, 
(SM, section 7). The blue shaded area represents mean ± 1 SD uncertainty in the measurements determining the process map. The yellow data point represents the directly measured LE between the two photons in a three-qubit string. The orange and purple data points describe a lower bound on the LE between the DE and the first emitted photon in a two- and a three-qubit string, respectively. The color-coded inset describes the configuration of qubits for each measured data point. To obtain the orange (purple) data points, we project the DE (second photon) on the state 
(
).
Next, we used the measured correlations to directly obtain the LE between the different qubits. First, we consider the entanglement between the DE and the emitted photon after one application of the cycle. Using the first set of DE-photon correlation measurements, we obtain a lower bound of 
for the negativity of their density matrix. The obtained lower bound is marked as an orange data point in Fig. 3. Second, we consider the three-qubit state obtained after two applications of the protocol (comprising the DE and the two emitted photons). We measure the density matrix of the two emitted photons, while projecting the DE on the state 
. The negativity of this measured density matrix, 
, is marked by the yellow data point in Fig. 3. The measured density matrix has fidelity of 
, with the maximally entangled state expected from the ideal case as given by Eq. 3. Note the good agreement between the LE obtained directly and the one obtained using the process map. Third, a lower bound on the LE between the DE and the first emitted photon, when the second emitted photon is projected on the optimized state 
, is extracted from our three-photon correlations. The value of this lower bound, 
, is marked in purple in Fig. 3. The full set of measurements leading to the above values of 
is given in the SM, section 8.
Finally, we use the DE-photon-photon correlation measurements to directly verify that the three-qubit state generated by our device exhibits genuine three-qubit entanglement (26). Because we did not project the DE spin on the 
states, the density matrix of the three qubits cannot be fully reconstructed (23–25). However, our measurements are sufficient to obtain bounds for the fidelity between the three-qubit state 
produced by our device and the three-qubit pure cluster state 
of Eq. 2, expected from the ideal process map. The value 
that we obtain is 
. The threshold for genuine three-qubit entanglement (35) is 
. The experimentally measured lower bound on 
is larger by more than 1 SD of the experimental uncertainty from this threshold. For full details on the calculation of 
, see the SM, section 9.
Both the quality of the state produced by our device and the correlation measurements can be improved. Our demonstration and analysis are based on high-repetition-rate (76 MHz) deterministic writing of the DE entangler and on time-tagged three-photon correlation measurements. Direct measurements of higher multiphoton correlations are very challenging in the current experimental setup, due to its limited light harvesting efficiency, where only about 1 in 700 photons are detected (SM, section 4). The limitation is mainly due to the low quantum efficiency of single-photon silicon avalanche photodetectors (
) and the low light collection efficiency from the planar microcavity (
). Both can, in principle, be considerably improved [see (36) and (20), respectively]. In addition, the current length scale of the LE is mainly limited by the radiative lifetime of the BiE. This lifetime can be considerably shortened by designing an optical cavity to increase the Purcell factor of the device (20) for this particular transition only.
Concluding remarks
We provide an experimental demonstration of a prototype device for generating on-demand 1D photonic cluster states (22). The device is based on a semiconductor QD and uses the spin of the DE as an entangler. Our prototype device can produce strings of a few hundred photons in which the localizable entanglement persists over five sequential photons. Further feasible optimizations of our device can enable faster and longer on-demand generation of higher-fidelity cluster states. Additionally, by using two nonidentical coupled QDs, it is possible to generate a 2D cluster state in a similar and scalable manner (37). A 2D cluster state carries a promise for robust implementation of measurement-based quantum computation (38). These may propel substantial technological advances, possibly bringing practical, widespread implementations of quantum information processing closer.
Supplementary Materials
www.sciencemag.org/content/354/6311/434/suppl/DC1
Materials and Methods
Figs. S1 to S9
Tables S1 and S2
References and Notes
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- Acknowledgments: We are grateful to P. Petroff for the sample growth and to T. Rudolph and J. Avron for useful discussions. The support of the Israeli Science Foundation (ISF), the Technion’s RBNI, and the Israeli Nanotechnology Focal Technology Area on Nanophotonics for Detection is gratefully acknowledged. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No 695188). The authors declare that they have no competing financial interests. The relevant data appear in this Research Article and in its supplementary materials.
