Quantifiers of Noise Reducibility Under Restricted Control

TL;DR

This work develops a principled, monotone framework for quantifying the usefulness of multitime quantum processes under restricted control. By defining three temporal-resource monotones $\overline{I}_{\hat{m}}, \overline{M}_{\hat{m}}, \overline{N}_{\hat{m}}$ based on relative entropy, the authors capture total, Markovian, and non-Markovian temporal correlations at chosen resolutions, and prove their monotonicity under IQI operations and coarse-graining. They establish subadditivity relations, analyze how these quantities behave under sequential and parallel composition, and relate them to generalised comb divergences, including reachable divergences that reflect operational constraints. The framework is instantiated in the dynamical-decoupling setting, showing that DD acts as temporal resource distillation and that existing DD schemes may underutilize available temporal correlations, with MODD providing a practical approximate method to approach the optimal quantifier. Overall, the paper provides robust, physically meaningful tools to quantify and optimize noise-reduction capabilities of quantum processes under realistic control limitations, with implications for the design of DD protocols and the study of non-Markovian quantum memory.

Abstract

The correlation structure of multitime quantum processes - succinctly described by quantum combs - is an important resource for many quantum information protocols and control tasks. Inspired by approaches for quantum states, we introduce quantifiers of the practical utility of quantum processes that satisfy monotonicity properties, thus overcoming shortcomings in previous state-motivated approaches. Applying these quantifiers to the problem of noise reduction of a quantum process under open-loop control, they are shown to represent the largest amount of temporal mutual information that a process can possibly exhibit. In addition, we study their resource composition behaviour and connect them to the recently introduced notion of generalised comb divergences. Finally, in light of these new quantifiers, we re-interpret the numerical findings of npj Quantum Information 9, 104 (2023) on the relationship of dynamical decoupling and non-Markovian memory, which were based on insufficient resource quantifiers, and show that its main conclusion - the interpretation of dynamical decoupling as a resource distillation - still holds.

Figures (5)

• Figure 1: Multi-time quantum process. Throughout, we consider the dynamics of a system $s$ coupled to an environment $e$ initially in state $\rho_\texttt{i}^e$. The system can be accessed/manipulated (depicted here by the maps $\mathcal{A}_1, \mathcal{A}_2, \dots, \mathcal{A}_n$) at times $\hat{n} =\{t_1, \dots, t_n\}$ in addition to the initial and final times $t_{\text{i}}$ and $t_{\text{f}}$, respectively. The evolution of the system and environment in between interventions is given by $\mathcal{T}$, and can differ for different time steps. Through the interaction with the environment, the process can display complex, non-Markovian correlations in time. The resulting process tensor description of the process is depicted by the grey outline. In this work, we are predominantly concerned with the IQI scenario (see Sec. \ref{['sec:restrictedcombs']}), where all $\mathcal{A}_k$ are independent. More generally, they can be correlated or consist of different layers of operations that map between different levels of temporal resolution (see Figs. \ref{['fig:superprocess']} and \ref{['fig:processtensorsuperprocess']}).
• Figure 2: Open System Dynamics and Choi states. Throughout, we consider the scenario of a system $s$ of interest that interacts with an environment $e$ and can be probed at times $t_\text{i}, t_1, \dots, t_n, t_\text{f}$. To map the process tensor (grey outline) that fully describes this multi-time setup onto a quantum state, half of a maximally entangled state is 'fed in' (represented using circuit notation by the SWAP operator) at each of the corresponding times, resulting in the $2n+2$-partite state $\mathbf{T}_{\hat{n}}$ of Eq. \ref{['eq:quantumprocessdef']}.
• Figure 3: Action of a superprocess and coarse-graining. A superprocess $\mathbf{Z}_{\hat{n}\hat{n}}$ maps a process on times $\hat{n}$ to another process on times $\hat{n}$. Temporal coarse-graining -- depicted by the identity channel $\mathcal{I}$ -- at times $\hat{n}\setminus \hat{m}$ reduces the temporal resolution from $\hat{n}$ to $\hat{m}$. The resulting process $\mathbf{T}'_{\hat{m}}$ is defined on the times $\hat{m}$ and given by $\llbracket \mathbf{T}_{\hat{n}}|\mathbf{Z}_{\hat{n}\hat{n}}|\mathbf{I}_{\hat{n}\setminus \hat{m}}\rrbracket = \llbracket \mathbf{T}_{\hat{n}}|\mathbf{Z}_{\hat{n}\hat{m}} \rrbracket$. Here, $\hat{n} = \{t_\text{i},t_1, t_2, t_3, t_\text{f}\}$ and $\hat{m}= \{t_\text{i},t_2, t_\text{f}\}$.
• Figure 4: Noise reduction in IQI. The process tensor $\mathbf{T}_{\hat{n}}$ is accessible at intermediate times $\hat{n}$, plus initial and final times $t_{\text{i}},t_{\text{f}}$. The superprocess $\mathbf{Z}_{\hat{n}\hat{n}}$ corresponds to experimental control, and transforms the process tensor $\mathbf{T}_{\hat{n}}$ on $\hat{n}$ times to another process $\mathbf{T}_{\hat{n}}'$ on the same set of times. In the operational scenario of IQI experimental control is memoryless, implying that $\mathbf{Z}_{\hat{n}\hat{n}}$ is realised by predetermined sequences of pre- and post-processing channels $\mathcal{V}_{t_{k}}: \mathcal{L}(\mathcal{H}_{\text{out}'_k}) \rightarrow \mathcal{L}(\mathcal{H}_{\text{out}_k})$ and $\mathcal{W}_{t_{k+1}}: \mathcal{L}(\mathcal{H}_{\text{in}_{k+1}}) \rightarrow \mathcal{L}(\mathcal{H}_{\text{in}'_{k+1}})$. Note that, in principle, $\mathcal{V}_{t_{k}}$ and $\mathcal{W}_{t_{k+1}}$ can change the dimension of the involved spaces, i.e., we can have $\mathcal{H}_{\text{in}_k} \ncong \mathcal{H}_{\text{in}'_k}$ and $\mathcal{H}_{\text{out}_k} \ncong \mathcal{H}_{\text{out}_k'}$. Temporal coarse-graining $\mathbf{I}_{\hat{n} \setminus \hat{m}}$ removes times from $\hat{n}$ that are not in $\hat{m}$. The resource quantifiers $\overline{I}_{\hat{m}},\overline{M}_{\hat{m}},\overline{N}_{\hat{m}}$ are concerned with the temporal correlations of the process at the coarse-grained level $\hat{m}$ of access. As depicted here, $\hat{m}$ is the empty set $\emptyset$, meaning that the resultant process after coarse-graining has no intermediate times. The relevant quantifier $\overline{I}_{\emptyset}=\overline{M}_{\emptyset}$ then becomes the maximum mutual information of the resultant channel that one can obtain under superprocesses $\mathbf{Z}_{\hat{n}\hat{n}}$ from IQI. If this mutual information of the resulting channel is larger than that of the one one would have obtained from performing no operations (i.e., only temporal coarse-graining), then the underlying resources within $\mathbf{T}_{\hat{n}}$ have successfully be employed for denoising.
• Figure 5: Composition of quantum processes. (a) Composing two processes $\mathbf{T}_{\hat{n}}$ and $\mathbf{S}_{\hat{n}'}$ in sequence yields a new time slot (denoted by $t'$) such that $\mathbf{S}_{\hat{n}'} \circ \mathbf{T}_{\hat{n}}$ is defined on $\hat{n} \cup \{t'\} \cup \hat{n}'$. (b) Composing two processes $\mathbf{T}_{\hat{n}}^A$ and $\mathbf{S}_{\hat{n}}^B$ in parallel yields a new process $\mathbf{T}_{\hat{n}}^A \otimes \mathbf{S}_{\hat{n}}^B$ on the same set of times $\hat{n}$.

Theorems & Definitions (12)

• Theorem 1: Irreversibility Resource Quantifiers
• proof
• Theorem 2: Subadditivity of $\overline{M}$ and $\overline{N}$
• proof
• Proposition 1: Sequential Composition
• proof
• Proposition 2: Superadditivity Under Parallel Composition
• proof
• Proposition 3: Invariance for fully uncorrelated Processes
• proof