Adversarial Information Gain in Non-ideal Quantum Measurements

Abstract

Performing a quantum measurement yields two different results: a classical outcome drawn from a probability distribution, according to Born's rule, and a quantum outcome corresponding to the post-measurement state. Quantum devices that provide both outcomes can be described through quantum instruments. In a realistic scenario, one can expect that the observer's obtained classical and quantum outcomes are non-ideal: this can be due to experimental limitations, but could also be explained by adversarial interference, that is, a second party that disturbs the device through a concealed measurement to obtain information. The second scenario can be interpreted through quantum compatibility, as it implies that both the observer's instrument and the adversary's measurement can be performed simultaneously. In this work, we show how the noise of the observer's device relates to the amount of information that the adversary can obtain. We study scenarios in which the adversary aims to acquire information on the same basis as the observer's measurement, or on a mutually unbiased basis with respect to the observer's basis. In both cases, we derive necessary and sufficient conditions for the compatibility of a single qubit non-ideal quantum instrument and a noisy meter, from which we obtain the maximum amount of information that the adversary can extract in terms of the noise parameters of the observer's instrument. Finally, we provide the device implementation from the adversary's point of view for the same basis scenario.

Figures (6)

• Figure 1: Device that performs a non-ideal quantum measurement on state $\rho$. The outputs correspond to the classical measurement outcome $x$ and the quantum post-measurement state $\tilde{\rho}$, which deviate from an ideal measurement due to noise.
• Figure 2: Observer and adversary viewpoints. In the top figure, the quantum instrument $\mathcal{I}$ corresponds to a black-box: the observer has full knowledge of the outputs of the measurement process for any given state, but possesses no information about the underlying mechanisms by which these transformations are performed. In the bottom figure, the full joint instrument $\mathcal{G}$ is shown: the adversary obtains the measurement outcome $y$ from meter $\mathsf{A}$. By taking the marginals over the adversary's outcome, the observer's expected outcomes are obtained.
• Figure 3: Regions of compatibility for instrument $\mathcal{I}^{\lambda, t, \hat{m}}$ and meter $\mathsf{A}^{s, \hat{n}}$ for different noise values $\lambda$, in the aligned (left) and complementary (right) directions.
• Figure 4: Adversary's maximal information gain $s_{\max}$ as a function of the sharpness $t$ for eleven equispaced values of $\lambda$ from 0 to 1.
• Figure 5: Adversary's implementation of the device: the adversary first retrieves information via the Lüders instrument $\mathcal{L}^{\mathsf{A}}$ of the meter $\mathsf{A}$, and performs a postprocessing instrument $\mathcal{R}^{(y)}$ according to the measurement outcome $y$. As the observer lacks knowledge of the adversary's outcome, summing over all possible outcomes $y$ returns the observer's expected classical and quantum outcomes.

Theorems & Definitions (11)

• Example 1: Noisy meters
• Example 2: Measure-and-prepare instruments
• Example 3: Lüders instrument
• Definition 1
• Theorem 1
• Theorem 2
• Corollary 1
• Proposition 1
• proof
• proof : Proof Theorem \ref{['thm:aligned']}