Nature Physics  Letter
Demonstration of entanglementbymeasurement of solidstate qubits
 Wolfgang Pfaff^{1}^{, }
 Tim H. Taminiau^{1}^{, }
 Lucio Robledo^{1}^{, }
 Hannes Bernien^{1}^{, }
 Matthew Markham^{2}^{, }
 Daniel J. Twitchen^{2}^{, }
 Ronald Hanson^{1}^{, }
 Journal name:
 Nature Physics
 Volume:
 9,
 Pages:
 29–33
 Year published:
 DOI:
 doi:10.1038/nphys2444
 Received
 Accepted
 Published online
Projective measurements are a powerful tool for manipulating quantum states^{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}. In particular, a set of qubits can be entangled by measuring a joint property^{3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13} such as qubit parity. These joint measurements do not require a direct interaction between qubits and therefore provide a unique resource for quantum information processing with wellisolated qubits. Numerous schemes for entanglementbymeasurement of solidstate qubits have been proposed^{8, 9, 10, 11, 12, 13}, but the demanding experimental requirements have so far hindered implementations. Here we realize a twoqubit parity measurement on nuclear spins localized near a nitrogenvacancy centre in diamond by exploiting an electron spin as a readout ancilla. The measurement enables us to project the initially uncorrelated nuclear spins into maximally entangled states. By combining this entanglement with singleshot readout we demonstrate the first violation of Bell’s inequality with solidstate spins. These results introduce a new class of experiments in which projective measurements create, protect and manipulate entanglement between solidstate qubits.
Subject terms:
At a glance
Figures
Main
A quantum measurement not only extracts information from a system but also modifies its state: the system is projected into an eigenstate of the measurement operator. Such projective measurements can be used to control and entangle qubits^{1, 2, 3, 4, 5, 6, 7, 8, 9}. Of particular importance is the qubit parity measurement that can create maximally entangled states^{10, 11, 12, 13}, plays a central role in quantum error correction^{14} and enables deterministic twoqubit gates^{8, 10}.
Qubit parity is a joint property that indicates whether an even or odd number of qubits is in a particular eigenstate. For two qubits, an ideal parity measurement projects the qubits either into the subspace in which the qubits have the same value (even parity) or into the subspace in which they have opposite values (odd parity). Crucially, only information on the parity is extracted: the measurement must not reveal any other information about the state of the qubits or otherwise disturb it.
To realize such a parity measurement the use of an ancillary system is required, as illustrated in Fig. 1a. First, both qubits are made to interact with this system such that its state becomes correlated with the parity of the joint twoqubit state. A subsequent highfidelity readout of the ancillary system then projects the two qubits into the even or odd parity subspace. For suitably chosen initial states, the parity measurement projects the qubits into a maximally entangled state. Implementations have been proposed for various qubit systems^{8, 10, 11, 12, 13}, but owing to the high demands on qubit control and ancilla readout experimental realization has remained elusive.
We achieve a heralded qubit parity measurement on two nuclear spins in diamond by using the electron spin of a nitrogenvacancy defect centre as a readout ancilla. We then use this parity measurement to project the two spins into any selected Bell state and verify the result by correlation measurements and quantum state tomography. Finally, we combine the measurementbased entanglement with a new twoqubit singleshot readout to demonstrate a violation of Bell’s inequality. As we eliminate all postselection and do not assume fair sampling, this experiment firmly proves that our parity measurement generates highpurity entangled states of nuclear spins.
Our implementation capitalizes on the excellent spin control in diamond developed over the past years^{15, 16, 17, 18, 19} and on the recently attained singleshot electron spin readout^{20}. We use the nitrogen nuclear spin N (^{14}N, I = 1, 0:m_{I} = 0, 1:m_{I} = +1) associated with a nitrogenvacancy centre and a nearby carbon nuclear spin C (^{13}C, I = 1/2, 0:m_{I} = +1/2, 1:m_{I} = −1/2) as qubits (Fig. 1b). By working at a low temperature (T<10 K) we can perform efficient initialization and projective singleshot readout of the ancillary electron spin (S = 1) by spinresolved resonant optical excitation^{20}. As the electron spin is coupled to the nuclear spin qubits through hyperfine interaction (Fig. 1c), the nuclear spins can be initialized and read out by mapping their state onto the electron spin^{15, 16, 17, 18, 20}.
The two nuclear spins are excellent qubits. They can be individually controlled with high precision in tens of microseconds by radiofrequency pulses (Fig. 1d,e) and they are well isolated from the environment: the dephasing time T_{2}^{*} exceeds a millisecond for both qubits. However, the direct interaction between the qubits (Fig. 1f) is weak compared with T_{2}^{*} and negligible on the timescales of our experiments (~100 μs).
The implementation of the parity measurement is depicted in the circuit diagram of Fig. 2a. First, the ancilla is initialized into 1_{a}. Then we apply two NOT gates on the ancilla controlled by the two qubits (Toffoli gates) through selective microwave pulses on the electron transitions for nuclear spin states CN = 00 and 11 (Fig. 1c). As a result, the ancilla is flipped to 0_{a} if the qubits are in a state of even parity, that is, if both qubits have the value 0 (first gate) or if both have the value 1 (second gate). This operation correlates the ancilla and the parity of the twoqubit state. Finally the ancilla is read out, which projects the qubits into either the even or oddparity subspace and yields the corresponding measurement outcome.
For the parity measurement to be useful in quantum protocols, it is required to be nondestructive so that the postmeasurement state can be used for further processing. The key experimental challenge is to preserve the phase of all possible twoqubit states of definite parity, such as the Bell states, during ancilla readout. We note that this demanding requirement is not present for measurements in which the ancilla readout projects all qubits onto eigenstates^{17, 19, 20}. Possible sources of nuclear qubit dephasing during prolonged optical readout are uncontrolled flips of the electron spin in the excited state^{20} and differences in hyperfine strength between the electronic ground and excited state^{21}. To avoid such dephasing, we use a short ancilla readout time. By conditioning on detection of at least one photon (outcome 0_{a}), we obtain a parity measurement that is highly nondestructive (measured ancilla postreadout state fidelity of (99±1)%) at the cost of a lower success probability (~ 3%).
Our parity measurement is ideally suited to generate highfidelity entangled states that are heralded by the measurement outcome. To also enable deterministic quantum gates, the measurement must yield a highfidelity singleshot outcome for both measurement results 0_{a} and 1_{a}. This can be achieved by further improvement of the electron readout, for instance by increasing the photon collection efficiency^{22} or by reducing electron spin flips through proper tuning of electronic levels^{23}.
We apply the parity measurement to project the two nuclear spin qubits into a Bell state. Figure 2b shows the circuit diagram of the protocol. We track the twoqubit state evolution by performing quantum state tomography at three stages. First, the qubits are initialized by projective measurement into 00 (Fig. 2c). The ancilla is reset to 1_{a} to be reused in the subsequent parity measurement and final readout. Then, we create a maximal superposition by applying a π/2pulse to each of the qubits, so that the twoqubit state is CN = (00+01+10+11)/2 (Fig. 2d). This state contains no twoqubit correlations. Finally, the parity measurement projects the nuclear spins into the even subspace, and thus into the Bell state . We find a fidelity with the ideal state of F = Φ^{+}ρ_{m}Φ^{+} = (90±3)%, where ρ_{m} is the measured density matrix (Fig. 2e). The deviation from a perfect Bell state can be fully explained by imperfect microwave πpulses that reset the ancilla to 1_{a} after the projection steps. The high fidelity of the output state is consistent with the nondestructive nature of the parity measurement.
For a maximally entangled state, a measurement of a single qubit yields a random result, whereas twoqubit correlations are maximal. We access these correlations through twoqubit measurements in different bases (Fig. 3a). The π/2 pulses implementing the basis rotations effectively transform the original state into other Bell states, which results in oscillations in the parity (Fig. 3b). In contrast, the singlequbit outcomes are found to be random, independent of the measurement basis (Fig. 3b).
The parity measurement can also project the qubits directly into each of the other Bell states. We create the states and by projecting into the odd subspace (Fig. 3c). The phase of the resulting state is preset deterministically by adjusting the phase of the pulses that create the initial superposition. We characterize the states Ψ^{+} and Ψ^{−} by correlation measurements in different bases (Fig. 3d,e). The visibility yields a lower bound for the state fidelity of (91±1)% and (90±1)%, respectively (Supplementary Methods). These results are consistent with the value obtained from quantum state tomography (Fig. 2e) and confirm the universal nature of our scheme. Furthermore, our results indicate how the parity measurement could be used to implement a nondestructive Bellstate analyser^{8, 13}. Although Ψ^{+} and Ψ^{−} show identical oddparity correlations in the Zbasis (Fig. 3d), they can be distinguished by a second parity measurement after a basis rotation (Fig. 3e).
Finally, we use our measurementbased scheme to observe a violation of Bell’s inequality with spins in a solid. This experiment places high demands on both the fidelity of the entangled state and on its readout^{24}, and therefore provides a pertinent benchmark for quantum computing implementations. We adapt the readout protocol to obtain a measurement of the complete twoqubit state in a single shot (Fig. 4a) and therefore do not rely on a fairsampling assumption^{24, 25}. To fully eliminate the need for postselection, we confirm before each experimental run that the nitrogenvacancy centre is in its negative charge state^{19} and that the optical transitions are resonant with the readout and pump laser^{20}.
We project into each of the four Bell states Φ^{±} and Ψ^{±}and measure the correlation function E(φ,θ) = P_{φ,θ}(00)+P_{φ,θ}(11)−P_{φ,θ}(01)−P_{φ,θ}(10) for all combinations of the Bell angles φ_{1,2} = {π/4, 3π/4} and θ_{1,2} = {0, π/2}. P_{φ,θ}(X) is the probability to measure state X after a rotation of the ^{13}C and ^{14}N qubits around the −Y axis by angles φ and θ, respectively (Fig. 4b). Figure 4c shows the resulting data for Φ^{−} and Ψ^{+}. We determine S = E(φ_{1},θ_{1})−E(φ_{1},θ_{2})−E(φ_{2},θ_{1})−E(φ_{2},θ_{2}) and observe a violation of the CHSH (Clauser–Horne–Shimony–Holt) inequality, S≤2, by more than 4 s.d. for each of the four Bell states, with a mean of S = 2.30±0.05 (Fig. 4d). The obtained values for S are lower than the theoretical maximum of owing to finite fidelities of the prepared Bell state and the singleshot readout. The main errors arise from imperfect microwave πpulses. On the basis of the separate characterization of a created Bell state (Fig. 2e) and of the readout (Fig. 4a) we expect S = 2.31±0.09, in agreement with the experimental result. For a perfect readout, a value of S = 2.5±0.1 would be obtained.
This violation of Bell’s inequality without assuming fair sampling demonstrates that our heralded parity measurement creates highpurity entangled states that can be used as input for deterministic quantum protocols such as deterministic teleportation. In contrast, early pioneering experiments with solidstate nuclear spins considered a subset of the full state and generated pseudopure states that contain no entanglement^{16, 26}. Therefore, our work constitutes the first unambiguous demonstration of entanglement between nuclear spins in a solid.
We have generated entanglement between two nuclear spins in diamond through a qubit parity measurement. Our scheme does not require a direct interaction between qubits and uses the fast but more fragile electron spin exclusively as an ancilla for the measurement. The protocol can be directly applied to other hybrid electron–nuclear systems such as phosphorous donors in silicon^{27, 28}. The generation of entanglement within a local register can be supplemented with remote entanglement through optical channels^{7, 29, 30, 31} to enable scalable quantum networks. Moreover, the presented parity measurements are a primary building block for deterministic measurementbased controlled NOT gates^{8} and quantum error correction^{14}. Therefore, our results mark an important step towards quantum computation in the solid state based on entangling, manipulating and protecting qubits by measurement.
Methods
Sample and setup.
We use a naturally occurring nitrogenvacancy centre in highpurity type IIa chemicalvapourdepositiongrown diamond with a 111 crystal orientation. Details of the experimental setup are given in ref. 20. All experiments are performed at temperatures between 8.7 and 8.85 K and with an applied magnetic field of ~ 5 G. We determine the following dephasing times (T_{2}^{*}) by Ramseytype measurements: (1.1±0.02) μs for the electron spin, (3.0±0.2) ms for the ^{13}C nuclear spin and (11.0±0.7) ms for the ^{14}N nuclear spin. The spectrum of the m_{s} = 0 to m_{s} = −1 transitions in Fig. 1c is measured by electron spin resonance with offresonant optical excitation.
Nuclear spin initialization and readout.
The nuclear spins are initialized by measurement. First the ancilla electron is initialized by depleting 0_{a} by optical pumping. Second the ancilla is flipped conditionally on state 00 and read out. Successful initialization in 0_{a}00 is heralded by a measurement outcome 0_{a}. The twoqubit state is read out in two steps. First we initialize the ancilla in 1_{a}. Second we sequentially probe each twoqubit eigenstate by flipping the ancilla conditional on the state probed and then reading out the ancilla. The result of the twoqubit readout is given by the first eigenstate for which the ancilla readout outcome is 0_{a}. In the CHSH experiments we obtain singleshot readout by repeating the probing sequence until ancilla outcome 0_{a} is obtained for one of the probed states.
For further details on the methods, characterization and error analysis see the Supplementary Information.
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Acknowledgements
We thank D. D. Awschalom, L. Childress, L. DiCarlo, V. V. Dobrovitski, G. D. Fuchs and J. J. L. Morton for helpful discussions and comments, and R. N. Schouten for technical assistance. We acknowledge support from the Dutch Organization for Fundamental Research on Matter (FOM), the Netherlands Organization for Scientific Research (NWO), the DARPA QuEST and QuASAR programmes, and the EU SOLID and DIAMANT programmes.
Author information
Affiliations

Kavli Institute of Nanoscience Delft, Delft University of Technology, PO Box 5046, 2600 GA Delft, The Netherlands
 Wolfgang Pfaff,
 Tim H. Taminiau,
 Lucio Robledo,
 Hannes Bernien &
 Ronald Hanson

Element Six, Ltd, Kings Ride Park, Ascot SL5 8BP, UK
 Matthew Markham &
 Daniel J. Twitchen
Contributions
W.P., T.H.T. and L.R. carried out the experiment. W.P. and T.H.T. analysed the data. H.B. fabricated the sample. M.M. and D.J.T. grew the diamond. W.P., T.H.T. and R.H. wrote the manuscript. All authors commented on the manuscript. R.H. supervised the project.
Competing financial interests
The authors declare no competing financial interests.
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Wolfgang Pfaff
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Tim H. Taminiau
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Lucio Robledo
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Hannes Bernien
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Matthew Markham
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Daniel J. Twitchen
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Ronald Hanson
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Supplementary information
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Supplementary Information