## Abstract

Our study reported entanglement among single spins in diamond. Lovett and Benjamin argue that three of six described entangled states were not achieved. Here, we explain our choice of entangled states and discuss their importance for quantum information processing. We also show that the eigenstates discussed by Lovett and Benjamin, although formally entangled and routinely generated in our experiments, cannot be used to detect nonlocal correlations.

We recently reported on multipartite entanglement among single spins in diamond (*1*). In their comment, Lovett and Benjamin (*2*) claim that three of the six demonstrated entangled states are in fact not entangled, namely the odd-parity Bell states Ψ^{+} and Ψ^{-} and the *W* state. They support this contention by approximating the eigenbasis of our system and deriving that two of the nuclear spin eigenstates are already entangled [|01〉 and |10〉 in figure 1B (*1*)] and that we disentangled these states. Although it would have been easier for us to prepare the eigenstates |01〉 and |10〉, which Lovett and Benjamin call the entangled ones, we created the states and , which we believe are the correct choice to show entanglement in this system, as justified below.

We welcome the opportunity to address this conceptual problem because it led to some initial confusion in the field of quantum computing. Although this issue is now largely resolved, those not familiar with the experimental details of how quantum logic is performed in spin systems may not be aware of it. The resolution of this problem is also closely related to concepts such as decoherence-free subspaces and logical qubits in spin systems, which are now very important in the field of quantum computing.

Lovett and Benjamin (*2*) say that entanglement is a physical resource that exists between physical entities, which they assign to the nuclear spins. In their product spin basis, they claim that some eigenstates look like Bell states, namely and . Although this state can be formally written like a Bell state, there is doubt that it can be called such. This is because, at the heart of entanglement, there is the correlation between two physically uncoupled systems (*3*)—a requirement that is not fulfilled here. Rather, the eigenstate is a mixing of the product spin states. Similarly, it also does not make sense to call the spin states of ortho and para hydrogen entangled. If, however, the spins are later physically separated, for example by breaking the molecular bonds and allowing the hydrogen atoms to fly apart (or the ^{13}C nuclei in the case of diamond), so that their strong coupling is removed and product states become eigenstates, then there will be entanglement. In the same way, Berkley *et al*. prepared an eigenstate of two strongly coupled Josephson-junction qubits that was formally entangled (*4*). This arose in a commentary by Wójcik *et al*. (*3*), which noted that such entanglement cannot be used to observe quantum correlations. However, Wójcik *et al*. further pointed out that there would be entanglement if the strong coupling were removed and the two qubits could be measured independently, as shown by Pashkin *et al*. (*5*). Thus, although it is theoretically possible to construct operators revealing the correlations among two strongly coupled spins as Bell-like, experimentally the strong coupling usually is removed, that is, the Hamiltonian is changed so that product states are eigenstates, and only then potential correlations can be monitored.

A particularly useful study of qubits is given in a coupled quantum dot spin system in (*6*). Unlike the NV diamond system, the Hamiltonian in the quantum dot system can be changed far more extensively by applying bias voltages. In a later study, Petta *et al*. also considered a case where it was possible to go from a situation where spin Bell states were eigenstates to a case where spin product states were eigenstates and back again (*7*), as in the case of Pashkin (*5*). This paper (*1*) is a particularly good illustration of the importance of the Hamiltonian in determining the correct choice for the unentangled spin qubit states, that is, the eigenstates.

In this sense, it is useless for us to show that we can prepare the eigenstates |01〉 and |10〉 to prove alleged entanglement because we cannot decouple the single spins/qubits from each other to achieve entanglement of the single nuclear spins, as is possible for quantum dots or superconducting qubits. In our basis, we drove four selective radio frequency (rf) transitions between all four eigenstates |00〉↔|01〉, |00〉↔|10〉, |10〉↔|11〉, and |01〉↔|11〉 [see figure 1 B (*1*)]. In this way we encoded two qubits that we can manipulate. We created entanglement among the states |00〉, |01〉, |10〉, and |11〉 of the NV center's m_{S} = -1 level shown in figure 1B in (*1*). This entanglement has been proven, for example, by doing Ramsey fringe experiments (*8*). This is the preparation of Bell states, a subsequent free precession period, and a final readout of their phases. In figure 2B(iv) in (*1*), Ramsey fringes following a phase dependence characteristic for multiple quantum transitions are clearly visible, thus demonstrating entanglement among the eigenstates |01〉 and |10〉. In addition to the Ramsey fringe experiments, state tomography has been performed on all produced states that revealed the correlations that are characteristic for entangled states; the same holds for the *W* states prepared. As such, we have a system at hand that is useful for quantum information application. Qubit information can be encoded and read out, as well as the correlations for which Bell states are famous. This is not the case for the eigenstates |01〉 and |10〉.

Another point made by Lovett and Benjamin (*2*) is that we made the a priori assumption that the eigenstates can be assigned as follows . This is the standard introduction to spin states, because it is conceptually simple to understand; these are the eigenstates in the absence of spin coupling. However, it often happens that the Hamiltonian of the total spin system has eigenstates that are not product states of the individual spins involved. When the Hamiltonian of the actual experimental spin system is stated, this physical picture must be modified [as was done in the supporting online material (SOM) (*1*)], but the notation is still correct as long as we are always working with eigenstates. We regret that we did not make this point more clearly in the original text.

If Lovett and Benjamin's derivation of eigenstates ( and (i.e., singlet and triplet states) is correct, why were we able to drive the nuclear spin transitions shown in figure 1B in (*1*) at all? The resolution of this question is that their derivation is only an approximation, similar to the one we described in the SOM for (*1*). In their comment (*2*), the eigenstates were derived from a Hamiltonian that was commuting with an operator *J*_{z} = *S*_{z} + *I*_{z1} + *I*_{z2} (their definition). They assumed from the SOM that the hyperfine tensors for the interaction of the electron with each of the two nuclei was collinear with the symmetry axis of the NV center, which is in fact not the case (*9*), as referenced in the SOM but not clearly mentioned. The symmetry axis of the hyperfine tensor is collinear with the vacancy-^{13}C axis that is tilted against the NV symmetry axis by the tetrahedral angle 109.5° [figure 1A (*1*)]. Thus, their operator does not commute with the Hamiltonian, and the singlet and triplet descriptions for these states are not correct. The wrong axis treatment also causes the incorrect derivation of 23 MHz for the energy splitting. Nevertheless, as we have stated in our SOM, the eigenstates |01〉 and |10〉 indeed have some singlet and triplet character.

In conclusion, for the case of Bell states, we chose a set of basis states among which we encoded two qubits. There, we are able to drive four selective radio frequency transitions, allowing us to manipulate nuclear spin states, that is, to perform quantum gates. In this way, we have been able to create and read out those particular correlations for which Bell states are famous and what is demanded for quantum computation. That's why we think this is the right basis and that we have created the right Bell states. The same holds for the *W* state.