3. Applications of nuclear spin single shot readout
3.2. Distinguishing between non-orthogonal quantum statesquantum states
3.2.2. Experimental implementation
Here, we implement the generalized measurements introduced in section 3.2.1 on the
14N nuclear spin of the NV. We employ projective, single shot readout of the nuclear spin and charge state pre-selection (cf. sections 2.2, 3.1.1). The basic measurement sequence is shown in fig. 3.3a. The first step is the initialization of the 14N nuclear spin into mI = |0i by measurement pre-selection. Then, single shot charge state detection is performed to pre-select for NV−. After these two initialization steps, the spin opera-tions for state preparation and measurement are carried out. For the preparation of the states |ai,|bi in eq. (3.8), a 2θ pulse is applied onto the mI = |0i ↔ |−1i transition
3.2. Distinguishing between non-orthogonal quantum states
14N Single shot
init |0〉
Single shot charge state
detection
14N state preparation /
spin manipulation
14N single shot
readout
Preparation Measurement
|-1〉
| 1〉+
|0〉 |0〉
2θ, 0(π) (|b〉)|a〉 θ1
|a〉~
|b〉~
|?〉
π/2 π a
b
Figure 3.3.: Measurement sequence. a, Basic sequence with14N initialization and charge state pre-selection. b, Pulse sequence on the14N spin for optimal USD.
with phase 0 or π, creating state |ai or|bi, respectively. The measurement basis, which depends on the chosen generalized measurement protocol as described in section 3.2.1 is realized by rotating the state appropriately on both transitions mI=|0i ↔ |−1i and mI = |0i ↔ |+1i (see below), such that the final measurement in the spin eigenbasis mI = {|0i,|−1i,|+1i} corresponds to the desired measurement basis. This final pro-jective readout is carried out by subsequent measurements on the eigenstates. Due to the finite readout fidelity and possible spin flips during readout, multiple positive results or no positive results for the subsequent measurements can be obtained. For multiple positive results, we take the first one as the final result. If no positive result is obtained, the measurement is ignored, which leads to an effective imperfect detection efficiency.
The measurement basis for the minimum error measurement is given in eq. (3.9), and is independent of the angle θ. It is obtained by applying a π/2 pulse with phase 0 onto the mI = |0i ↔ |−1i transition. Additionally, we apply a π rotation onto the mI = |0i ↔ |+1i transition, such that the final measurement is performed in the mI ={|−1i,|+1i}, which yields higher fidelities than measurements on mI =|0i. The probability to obtain at least one positive measurement result (effective detection effi-ciency) for this implementation was ≈83.1%.
For the standard unambiguous state discrimination, the operations ˆUa = |0iha|+
|1iha⊥| and ˆUb =|0ihb|+|1ihb⊥| yield the two possible measurement bases given in eq.
(3.11). These operations are realized by 2θand -2θ pulses for ˆUaand ˆUb, respectively, on themI=|0i ↔ |−1itransition. Again, aπrotation onto themI =|0i ↔ |+1itransition is performed before readout. Depending on the chosen basis, mI = |−1i corresponds to |a⊥i or |b⊥i, and mI = |−1i corresponds to |ai or |bi. Here, the effective detection
efficiency was≈84.6%.
The pulse sequence for the optimal USD is shown in fig. 3.3b. The operation ˆU that generates the measurement basis defined in eq. (3.14) is (in the basis{|0i,|−1i,|+1i})
Uˆ = 1
√2
tanθ −1 −√
1−tan2θ
tanθ 1 −√
1−tan2θ
q2(1−tan2θ) 0 √ 2 tanθ
. (3.16)
We decompose this operation into two rotations [136, 137], which each act on only one spin transition. The decomposition is given by ˆU = ˆT0,−1Tˆ0,+1, with
Tˆ0,−1 = 1
√2
1 −1 0
1 1 0
0 0 √
2
, (3.17)
Tˆ0,+1 =
tanθ 0 −√
1−tan2θ
0 1 0
√1−tan2θ 0 tanθ
. (3.18)
Tˆ0,+1 corresponds to a θ1 = 2 arcsin(√
1−tan2θ) rotation on the mI = |0i ↔ |+1i transition and ˆT0,+1 to a π/2 rotation on the mI = |0i ↔ |−1i transition. As above, finally aπ rotation on themI =|0i ↔ |+1itransition is applied. This has the additional effect that if the pulse sequence is not successful due to e.g. imperfect electron spin initialization, the result will be inconclusive. For the final measurement, a positive result on mI = |0i, |−1i, and |+1i corresponds to |?i, |ai, and |bi, respectively, where
|?iindicates the inconclusive result. Here, the average detection efficiency was≈90.2%.
Method d unambiguous error % efficiency %
SUSD 2 yes ∼3.5 84.6
IDP 3 yes 4−7.5 90.2
Helstrom 2 no >3.5 83.1
Table 3.1.: Comparison of the three implemented measurement protocols. Columndis the needed dimension of the Hilbert space. The Helstrom measurement has an inherent error probability; here we show the minimum error due to measurement imperfections.
Results Table 3.1 compares the three implemented measurement protocols. Fig. 3.4 shows the measurement results. The probability of obtaining a correct results ispcorr = p(a|a)pa+p(b|b)pb, where p(i|j) is the conditional probability to get result i if state j was prepared, and pa, pb are the initial state probabilities for |ai, |bi, respectively.
Likewise, the probability for an incorrect result is perr=p(b|a)pa+p(a|b)pb, and for an inconclusive resultp? =p(?|a)pa+p(?|b)pb.
3.2. Distinguishing between non-orthogonal quantum states
standard USD min. error optimal USD
standard USD optimal USD standard USD
min. error optimal USD 0.0
0.2 0.4 0.6 0.8 1.0
pcorr
0.0 0.2 0.4 0.6 0.8 1.0
p?
0.0 0.2 0.4 0.6 0.8 1.0
perr
0.0 0.2 0.4 0.6 0.8 1.0
⟨a|b〉 0.0 0.2 0.4⟨a|b〉0.6 0.8 1.0 0.0 0.2 0.4⟨a|b〉0.6 0.8 1.0
a b c
Figure 3.4.: Measurement results. a, Probability for correct resultpcorr =p(a|a)pa+p(b|b)pb. b, Probability for inconclusive result p? = p(?|a)pa+p(?|b)pb. c, Probability for incorrect result perr=p(b|a)pa+p(a|b)pb. The solid / dotted lines show the expected result for ideal measurements.
A main drawback of the standard USD is that only one state can be probed, i.e. if the system is initially in the other state, the result will always be inconclusive. This can be seen in fig. 3.4a, b: The probability for a correct result is at most 0.5, and the probability for an inconclusive result always >0.5. For optimal USD, both states can be probed independently, leading to better results for pcorr and p?. However, due to the increased experimental complexity the error probability is slightly increased, see fig.
3.4c and table 3.1.
An important comparison is the probability for an incorrect result perr of the USD and the minimum error measurement. Ideally, for USD there should be no incorrect results, whereas these are expected for the minimum error measurement. Nevertheless, due to experimental imperfections there will be incorrect results for all measurement protocols. Fig. 3.4c shows that for small overlap ha|bi, perr is similar for all three implementations, and mainly limited by the readout fidelity for the minimum error measurement. Only for larger overlap ha|bi, perr, the intrinsicperr of the minimum error measurement dominates. On the other hand, the probability of a correct result for the minimum-error measurement is higher than for USD, as expected.
Conclusions Here, we have implemented generalized measurements for the first time in solid state qubits, which has previously only been done optically [138,139]. This was achieved by making use of the qutrit character of the 14N nuclear spin. The ability to perform these type of measurements is of interest for QIP, and confirms this potential application of the NV.