Power of one bit of quantum information in quantum metrology

We present a model of quantum metrology inspired by the computational model known as deterministic quantum computation with one quantum bit (DQC1). Using only one pure qubit together with $l$ fully mixed qubits we obtain measurement precision (defined as root-mean-square error for the parameter being estimated) at the standard quantum limit, which is typically obtained using the same number of uncorrelated qubits in fully pure states. In principle, the standard quantum limit can be exceeded using an additional qubit which adds only a small amount of purity. We show that the discord in the final state vanishes only in the limit of attaining infinite precision for the parameter being estimated.

Figure 1

(a) A DQC1-complete problem is to compute the normalized trace of a unitary transformation. After several runs of the circuit, an estimate is obtained for $\langle {\sigma }_x\rangle +i\langle {\sigma }_y\rangle =\text{tr}\left(U\right)/2^l$, with precision depending only on the number of runs. The protocol also works when the control qubit is partially pure at the start—as given by the state in Eq. (5). In this case, the number of runs must be increased by a factor $1/{\epsilon }^2$ to achieve the same precision as when the control qubit is initially pure. (b) For the general DQC1 problem, additional $n\sim log\left(l\right)$ pure qubits can be introduced without altering the computational power [5, 6].

Figure 2

General scheme for phase estimation: There are $n$ pure qubits, $m$ semipure qubits, and $l$ fully mixed qubits (${\mathbb{1}}_2=\mathbb{1}/2$). The first qubit is the control, and the remaining qubits constitute the register. A bulk cnot is used to prepare the probe, and each qubit is then subjected to the unitary operation given in Eq. (1) for which $\varphi$ is to be determined. The readout procedure is adaptive: ${\theta }_r$ is the estimate for $\varphi$ after the first $r-1$ rounds. This value is used to configure the readout circuit, which is a bulk cnot followed by a bulk controlled $v_r$, and measurement on the control.

Figure 3

$F$ computed from the probability distribution in Eq. (9) for $n=6$ and $m=l=0$ (dotted); $n=l=0$ and $m=11$ with $\epsilon =0.49$ (dashed); and $l=48$ and $n=m=0$ (solid). All three cases have the same value for $F_{\mathrm{q}}$. When there is no prior information about the value of $\varphi$, an initial estimate of its value can be obtained using a small number of measurements with single-qubit probes, half with an adaptive setting $\theta =0$ and half with $\theta =\pi /2$. A high-precision estimate can then be obtained using a larger number of measurements with probe states having register qubits. The adaptive procedure used at this stage would adjust the setup towards maximum sensitivity, while ensuring that estimates of $\varphi$ do not become multivalued.