Memory-built-in quantum cloning in a hybrid solid-state spin register

As a way to circumvent the quantum no-cloning theorem, approximate quantum cloning protocols have received wide attention with remarkable applications. Copying of quantum states to memory qubits provides an important strategy for eavesdropping in quantum cryptography. We report an experiment that realizes cloning of quantum states from an electron spin to a nuclear spin in a hybrid solid-state spin register with near-optimal fidelity. The nuclear spin provides an ideal memory qubit at room temperature, which stores the cloned quantum states for a millisecond under ambient conditions, exceeding the lifetime of the original quantum state carried by the electron spin by orders of magnitude. The realization of a cloning machine with built-in quantum memory provides a key step for application of quantum cloning in quantum information science.

Our experimental setup is described in detail in Ref. [1]. We use a home-built confocal microscopy, with an oilimmersed objective lens, to address and detect single NV center in a type IIa single-crystal synthetic diamond sample from the Element Six company. For each experimental cycle, we start the sequence with 2 µs laser illumination (at 532 nm wavelength) to polarize the NV electron and the nearby nuclear spins (C 13 and the host N 14 ) and end it with a 3 µs laser pulse for spin state detection. The polarization of the nuclear spins is achieved through the almost degeneracy of the electron spin levels |m = 0⟩ and |m = −1⟩ in the optically excited state under an external magnetic field of 427 G, which facilitates electron-spin nuclear-spin flip-flop process during optical pumping [2].
To read out, the spin states are distinguished through their different fluorescence levels under illumination of a short laser pulse, and the fluorescence levels of different spin states are calibrated through the method described in Ref. [1]. We collect signal photons for 0.3 µs right after the detection laser rises to its full intensity, and another 0.3 µs for reference 2 µs later. With a photon collection rate of 80 kHz, we have an average of 0.024 photon counts per cycle. For measurement of each datum, we repeat the experimental cycle at least 6 × 10 6 times, resulting in a total photon counts of 1.4 × 10 5 . The error bars of our data account for the statistical error associated with the photon detection. We use Monte Carlo simulation to calculate the error bar of each datum, assuming a Poissonian distribution for the photon counts.

ELECTRON SPIN DEPHASING AND SPIN ECHO
One step of our cloning operation is achieved with a r.f. pulse to perform a conditional gate on the nuclear spin. As the gate on the nuclear spin is slow with the gate time longer than the electron spin coherence time under free evolution (about 1.7 µs), we need to consider the influence of dephasing of the electron spin. The dephasing of the electron spin is caused by the hyperfine interaction with a bath of nuclear spins. The effect of the nuclear spin bath on the electron spin can be modelled by an effective magnetic field B (t), which is slowly fluctuating with time t. Within 100 µs, the time dependence of B (t) is negligible. So the field B just takes an unknown constant value for each experimental trial, but varies from trial to trial. The influence of the unknown B field needs to be removed in our measurements to verify quantum cloning, and this can be achieved with a single Hahn spin echo pulse, as we show now.
In our implementation, a r.f. π/2-pulse performs the controlled Hadamard gate CH 12 . Assume this gate takes time t g . With influence of the B field taken into account, the state after this gate is |0, ↑⟩ / √ 2 + e i(φ+Btg) |1⟩ (|↑⟩ + |↓⟩) /2, where we have assumed that the accumulated phase caused by the B field is s z Bt g (s z is the electron spin Zeeman number), neglecting the unimportant pre-factor. A microwave π-pulse, which is short compared with 1/B and considered to be instantaneous, applies to the transition |1, ↓⟩ ↔ |0, ↓⟩ and transfers the state to the form |0, ↑⟩ / √ 2 + e i(φ+Btg) (|0, ↓⟩ + |1, ↑⟩) /2. Then, we wait for a time t 1 and applies an instantaneous microwave π-pulse (spin echo pulse) to flip the electron spin |1⟩ ↔ |0⟩, and the state of the system becomes |1, ↑⟩ / √ 2 + e iφ ( e iBtg |1, ↓⟩ + e iB(tg+t1) |0, ↑⟩ ) /2. After another delay time t 2 , we perform the quantum state tomography measurements on the state |Φ f ⟩ = e iBt2 |1, ↑⟩ / √ 2 + e iφ ( e iB(tg+t2) |1, ↓⟩ + e iB(tg+t1) |0, ↑⟩ ) /2. Note that quantum state tomography consists of measurements in several complementary bases to map out the density matrix elements of the final state one by one [3]. In implementation of quantum state tomography, we need to apply a combination of r.f. and microwave pulses to rotate the measurement basis, where the r.f. pulse may take another time t d (note that t d is basis-dependent). As an example, if we measure the coherence |1, ↑⟩ ⟨1, ↓| between the first two terms of |Φ f ⟩, we apply a microwave π-pulse (instantaneous) to transfer |1, ↓⟩ to |0, ↓⟩, a r.f. π-pulse with time t d to transfer |1, ↑⟩ to |1, ↓⟩, and a microwave π/2-pulse to the transition |1, ↓⟩ ↔ |0, ↓⟩ to transfer the coherence term |1, ↑⟩ ⟨1, ↓| finally to the population difference in |1⟩ and |0⟩ levels, which can be read out through optical detection. With choice of t 1 = t 2 and t g = t d , the effect of the unknown B field is removed. Other coherence terms of the final density matrix are read out by similar methods with an appropriate choice of t 2 and t d .

PROOF OF THE FIDELITY BOUND FOR CLASSICAL CLONING
The fidelity bound for classical cloning is known to be 2/3 for university quantum cloning where the input state is randomly taken from arbitrary qubit states [4]. The fidelity bound becomes 3/4 for classical phase covariant quantum cloning with the input randomly taken from the states on the equator of the Bloch sphere. Motivated by the connection with the BB84 quantum key distribution protocol, we consider here that the input state is randomly taken from a finite set of states with φ = 0, π/2, π, 3π/2. We prove that the average fidelity for any classical cloning machine is still upper bounded by 3/4 in this case.
For classical cloning, we first make a measurement on the input state and then prepare two copies of the states based on the measurement information. The best detection strategy is given by a von Neumann measurement in the qubit space [4], which projects the input state |Ψ⟩ to |v⟩ and |v ⊥ ⟩ with probabilities |⟨v|Ψ⟩| 2 and |⟨v ⊥ |Ψ⟩| 2 , respectively, where |v⟩ and |v ⊥ ⟩ denote two arbitrary orthogonal qubit states. When the measurement outcome is |v⟩ (or |v ⊥ ⟩), we prepare two copies of the outcome state |v⟩ (or |v ⊥ ⟩), with the cloning fidelity |⟨v|Ψ⟩| 2 (or |⟨v ⊥ |Ψ⟩| 2 ) for this case. The average classical cloning fidelity for the input state |Ψ⟩ is given by When the input |Ψ⟩ is randomly taken from the set of four states 2 with equal probability p = 1/4, the best choice of the measurement basis vectors |v⟩ and |v ⊥ ⟩ apparently should also be from the states on the equator of the Bloch sphere with |v⟩ = ( |0⟩ where the optimal ϕ is to be determined. The average classical cloning fidelity is then The fidelity F c is actually independent of the choice of parameter ϕ. As F c is for the optimal classical cloning strategy, we conclude that the average fidelity for any classical cloning machine is always upper bounded by 3/4 when the input is taken from the set of four different states as in the BB84 protocol.