Entropic uncertainty under indefinite causal order and input-output direction

TL;DR

This work analyzes memory-assisted entropic uncertainty relations (MA-EUR) under Pauli-channel noise affecting the memory qubit, using higher-order quantum processes—the quantum switch and quantum time-flip—to create indefinite causal order and indefinite input-output direction. By modeling the memory as a Bell-diagonal state and evaluating $S(Q|B)+S(R|B)$ for $Q=\sigma_x$ and $R=\sigma_z$, the authors derive explicit expressions for the left-hand side and the MA-EUR bound under three regimes: single-use, self-switched, and time-flipped Pauli channels. They show that both switch and time-flip can reduce the total uncertainty relative to direct noise, with a clear threshold $p>1/2$ for self-switch advantages and a simple condition $|\tau_x|>|\lambda_x|$ for time-flip advantages, highlighting indefinite causal structures as practical resources for noise mitigation in MA-EUR tasks. The results imply that indefinite causal order and input-output direction can enhance robustness of MA-EUR-based protocols and related quantum information applications.

Abstract

Entropic uncertainty relations quantify the limits on the predictability of quantum measurements. When the measured system is correlated with a quantum memory, these limits are described by the memory-assisted entropic uncertainty relation (MA-EUR). We examine the behavior of MA-EUR when the memory qubit undergoes noisy dynamics implemented via high-order controlled processes, namely, the quantum switch and the quantum time-flip. We consider a setting in which the control qubit is the very system on which the measurements are performed, while the target qubit serves as a noisy quantum memory. Focusing on Pauli channels, we show that feeding them into the quantum switch and the quantum time-flip can significantly reduce the total entropic uncertainty as compared to their direct application. Our results reveal that indefinite causal order and input-output direction can serve as resources to mitigate the effects of noise in the context of MA-EUR and its applications.

Figures (5)

• Figure 1: Assuming that $\alpha_x=\alpha_y=0.5$ and $\alpha_z=0$, Bob's uncertainty about x-measurement (a) and z-measurement (b) for switched (sw) and single-use (su) Pauli channels in terms of the overall error probability $p$. (c) Bob's total uncertainty for switched $U_{\text{sw}}$ and single-use $U_{\text{su}}$ channels. (d) Total uncertainty $U_{\text{sw}}$ and its lower bound $B_{\text{sw}}$ for switched channels.
• Figure 2: Supposing $\alpha_x=0.5$, $\alpha_y=0.1$ and $\alpha_z=0.4$, Bob's uncertainty about x-measurement (a) and z-measurement (b) for switched (sw) and single-use (su) Pauli channels in terms of the overall error probability $p$. (c) Bob's total uncertainty for switched $U_{\text{sw}}$ and single-use $U_{\text{su}}$ channels. (d) Total uncertainty $U_{\text{sw}}$ and its lower bound $B_{\text{sw}}$ for switched channels.
• Figure 3: Density plots of $\Delta U = U_\text{sw}-U_\text{su}$, i.e., the difference between the total entropic uncertainties for the self-switched and the single-use Pauli channels. White regions in the panels correspond to the non-physical regime, where $\alpha_x+\alpha_y+\alpha_z=1$ with $\alpha_x,\alpha_y,\alpha_z\geq0$ is not satisfied. There exists no switch based advantage for the total uncertainty when $p=0.25$ (a) and $p=0.50$ (b) as $\Delta U$ is always positive for Pauli channels. The panels in (c) and (d) display the regions where $\Delta U<0$, and thus where the total entropic uncertainty of Bob about Alice's measurements is decreased due to the self-switching of Pauli channels, respectively, for $p=0.75$ and $p=1.00$. Thick dashed lines correspond to the contours on which $U_\text{sw}= U_\text{su}$.
• Figure 4: For $\alpha_x=0.5$, $\alpha_y=0.3$ and $\alpha_z=0.2$, Bob's uncertainty about x-measurement (a) and z-measurement (b) for time-flipped (tf) and single-use (su) Pauli channels in terms of the overall error probability $p$. (c) Bob's total uncertainty for time-flipped $U_{\text{tf}}$ and single-use $U_{\text{su}}$ channels. (d) Total uncertainty $U_{\text{tf}}$ and its lower bound $B_{\text{tf}}$ for time-flipped channels.
• Figure 5: Density plots of $\Delta U' = U_\text{tf}-U_\text{su}$, i.e., the difference between the total entropic uncertainties for the time-flipped and the single-use Pauli channels. White regions in the panels correspond to the non-physical regime, where $\alpha_x+\alpha_y+\alpha_z=1$ with $\alpha_x,\alpha_y,\alpha_z\geq0$ is not satisfied. There always exists time-flip based advantage (except for $\alpha_y=0$) for total uncertainty when $p=0.25$ (a) and $p=0.50$ (b) as $\Delta U'$ is always negative. The panels in (c) and (d) display the regions where $\Delta U'<0$, and thus where the total uncertainty of Bob about Alice's measurements is decreased due to the time-flipped Pauli channels, respectively, for the cases $p=0.75$ and $p=1.00$. Thick dashed lines correspond to the contours on which $U_\text{tf}= U_\text{su}$.