Quantum remote sensing with asymmetric information gain

Typically, the aim of quantum metrology is to sense target fields with high precision utilizing quantum properties. Unlike the typical aim, in this paper, we use quantum properties for adding a functionality to quantum sensors. More concretely, we propose a delegated quantum sensor (a client-server model) with security inbuilt. Suppose that a client wants to measure some target fields with high precision, but he/she does not have any high-precision sensor. This leads the client to delegate the sensing to a remote server who possesses a high-precision sensor. The client gives the server instructions about how to control the sensor. The server lets the sensor interact with the target fields in accordance with the instructions, and then sends the sensing measurement results to the client. In this case, since the server knows the control process and readout results of the sensor, the information of the target fields is available not only for the client but also for the server. We show that, by using an entanglement between the client and the server, an asymmetric information gain is possible so that only the client can obtain sufficient information on the target fields. In our scheme, the server generates the entanglement between a solid-state system (that can interact with the target fields) and a photon, and sends the photon to the client. On the other hand, the client is required to possess linear-optics elements only including wave plates, polarizing beam splitters, and single-photon detectors. Our scheme is feasible with the current technology, and our results pave the way for an application of quantum metrology.

Figure 1

A schematic diagram to illustrate client-server based quantum sensing. We consider the cases during and after the delegation of the sensing. (a) During the delegation, the client sends a sample to the server who holds a high-precision quantum sensor. The client gives the server instructions about how to control the quantum sensor for the measurement of the sample, and the server obeys the instructions. The server sends the measurement results to the client, and the server returns the sample to the client. (b) After the delegation, the server can try to estimate the information of the sample from the classical data remaining in the server's quantum sensor. Interestingly, our protocol described in this paper prevents the server from obtaining the information of the sample from the remaining data while the client can recover the information of the sample, which we call asymmetric information gain.

Figure 2

Comparison of the standard sensing protocol with our remote sensing protocol. Here, $|\pm \rangle \equiv \left(\right|0\rangle \pm |1\rangle )/\sqrt{2}$ and $M$ is the repetition number. (a) In the standard protocol, there are three steps such as the state preparation, the interaction, and the readout. (b) In our protocol, the state preparation is composed of the Bell pair generation (between the server and the client) and a subsequent measurement by the client, while the interaction and the readout are implemented at the server's side similar to the standard protocol. Since the server does not know the client's measurement result, the server cannot know which single-qubit state is prepared at the server's side before the interaction while the client can know it. This difference realizes the asymmetric information gain.

Figure 3

The quantum circuit representation of the standard quantum metrology protocol. Here, $\mathcal{H}$ represents the time evolution by the Hamiltonian in Eq. (1), $S^{†}\equiv |0\rangle \langle 0|-i|1\rangle \langle 1|, H$ is the Hadamard gate, and the meter symbol represents the ${\widehat{\sigma }}_z$-basis measurement.

Figure 4

The required number $N=8k$ of qubits as a function of $\epsilon$. Here, $k=⌈75ln(2/\delta )/\left[8{(\epsilon -3\mathrm{\Delta })}^2\right]⌉$. The bottom, middle, and top lines show the value of $8k$ for $(\delta ,\mathrm{\Delta })=({10}^{-3},0), ({10}^{-5},0)$, and $({10}^{-3},\epsilon /10)$, respectively.

Figure 5

(a) The quantum circuit representation of our delegated metrology where ${\rho }_{\mathrm{tgt}}$ denotes the state of the target register between the server and the client. It is worth mentioning that, by taking a large limit of the number $k, {\rho }_{\mathrm{tgt}}$ approaches the ideal Bell pair. (b) The quantum circuit equivalent to the circuit in (a). The vertical line represents the controlled-${\widehat{\sigma }}_z$ gate $|0\rangle \langle 0|\otimes I+|1\rangle \langle 1|\otimes {\widehat{\sigma }}_z$. These two circuits output $s\oplus o$ with the same probability.

Figure 6

The client's uncertainty $\delta {\omega }_C^{\left(U\right)}$ against the number $M$ of repetitions where we set $t=1$. Although the horizontal axis represents discrete values, we use continuous lines for plots as guides for the eyes.

Figure 7

Plot of $\delta {\omega }_S^{\left(L\right)}/\delta {\omega }_C^{\left(U\right)}$ against the number $N=8k$ of qubits required to extract a single two-qubit state that is close to the Bell pair by the random-sampling test where we set $\delta ={10}^{-6}$ and $\mathrm{\Delta }=0$. Although the horizontal axis represents discrete values, we use continuous lines for plots as guides for the eyes.

Figure 8

Plot of $\delta {\omega }_S^{\left(L\right)}/\delta {\omega }_C^{\left(U\right)}$ against the number $M$ of repetitions where we set $\delta ={10}^{-6}$ and $\mathrm{\Delta }=0$. Although the horizontal axis represents discrete values, we use continuous lines for plots as guides for the eyes.

Figure 9

Plot of $\mathrm{\Delta }{\omega }_S^{\left(L\right)}/\mathrm{\Delta }{\omega }_C^{\left(U\right)}$ against the number $N=8k$ of qubits required to extract a single two-qubit state that is close to the Bell pair by the random-sampling test where we set $\delta ={10}^{-6}, \mathrm{\Delta }=0$, and $\tilde{s}=2$. Although the horizontal axis represents discrete values, we use continuous lines for plots as guides for the eyes.

Figure 10

Plot of $\mathrm{\Delta }{\omega }_S^{\left(L\right)}/\mathrm{\Delta }{\omega }_C^{\left(U\right)}$ against the number $M$ of repetitions where we set $\delta ={10}^{-6}, \mathrm{\Delta }=0$, and $\tilde{s}=2$. Although the horizontal axis represents discrete values, we use continuous lines for plots as guides for the eyes.