Distinguishing between Nonorthogonal Quantum States of a Single Nuclear Spin

An important task for quantum-information processing is optimal discrimination between two nonorthogonal quantum states, which until now has been realized only optically. Here, we present and compare experimental realizations of optimal quantum measurements for distinguishing between two nonorthogonal quantum states encoded in a single ${}^{14}\mathrm{N}$ nuclear spin at a nitrogen-vacancy defect in diamond. Implemented measurement schemes are the minimum-error measurement (known as Helstrom measurement), unambiguous state discrimination using a standard projective measurement, and optimal unambiguous state discrimination [known as Ivanovic-Dieks-Peres (IDP) measurement], which utilizes a three-dimensional Hilbert space. This allows us to benchmark the IDP measurement against the standard projective measurements. Measurement efficiencies are found to be above 80% for all schemes and reach a value of 90% for the IDP measurement.

Figure 1

(a) Illustration of the applied radio frequency pulses to prepare the nonorthogonal states and perform the IDP measurement. For the Helstrom measurement, we omit the ${\theta }_1$ pulse. (b) Histogram of results for single-shot readout of $|\tilde{a}⟩$ for several preparations of a single ${\mathrm{NV}}^{-}$ center in the state $|a⟩$ with $\theta =\pi /4$. In each event (preparation and detection), a level of photocounts below the threshold is taken to correspond to the system having been prepared in the state $|a⟩$.

Figure 2

Geometrical representation of (a) the IDP measurement. The initial two-dimensional Hilbert space, where $|a⟩$ and $|b⟩$ live, is spanned by $\{|0⟩,|-1⟩\}$. The measurement is a projective measurement in the basis $\{|\tilde{a}⟩|\tilde{b}⟩,|?⟩\}$, where $|\tilde{a}⟩$ is orthogonal to $|b⟩$ and $|\tilde{b}⟩$ is orthogonal to $|a⟩$. It can be realized by first making the unitary operation $\widehat{U}$ in Eq. (4), followed by a projective measurement in the $\{|0⟩,|-1⟩,|+1⟩\}$ basis. (b) The Helstrom measurement basis $\{|a^{\prime }⟩,|b^{\prime }⟩\}$.

Figure 3

Experimental results of unambiguous and minimum-error discrimination of nonorthogonal states. The two nonorthogonal states $|a⟩$ and $|b⟩$ are prepared with equal prior probabilities $p_a=p_b=1/2$. The probabilities for correct identification $p_{\mathrm{corr}}=p(a|a)p_a+p(b|b)p_b$, inconclusive outcomes $p_{?}=p(?|a)p_a+p(?|b)p_b$, and incorrect outcomes $p_{\mathrm{err}}=p(b|a)p_a+p(a|b)p_b$ are shown in (a), (b), and (c), respectively. Probabilities are plotted as functions of overlap $⟨a|b⟩=\cos 2\theta$. The experimental error bars account only for uncertainty due to measurement shot noise.