Quantum memory precludes mixed-unitary dynamics

Abstract

Unital quantum channels, defined by their property of leaving the maximally mixed state invariant, form an important class of quantum operations. A distinguished subset of these channels can be represented as a probabilistic mixture of unitary evolutions. Characterizing channels that do not admit such a decomposition is in general a hard problem with significant implications for noise mitigation in quantum technologies and for fundamental problems in quantum information theory. Here we establish a link between mixed-unitarity of unital channels and the (quantum) nature of the memory effects in non-Markovian dynamics. Translating the problem into the language of process tensors, this connection yields a hierarchy of semidefinite programs that provides numerically efficient witnesses for non-mixed-unitary behavior, outperforming existing criteria. We demonstrate the power of this approach through illustrative examples of unital channels in dimensions three and four.

Figures (5)

• Figure 1: Geometric representation of the space of quantum channels with the five unital channels investigated thoroughly in this Letter being highlighted. Two of them are mixed-unitary (MU) and the other three channels are non-mixed-unitary (non-MU). Note that this two-dimensional depiction of the high-dimensional space of quantum channels does not capture all relevant properties, although convexity properties are reflected correctly.
• Figure 2: Witnesses of non-mixed-unitaryness in the family of CPT maps given by Eq. \ref{['eq:family_ls_id']}. The blue solid line represents the quantum-memory-based witness $s_{\text{QM}}$ and indicates non-mixed-unitaryness if it is negative. The orange dashed line computed the witness $s_\mathrm{MW}$ according Eq. \ref{['eq:mendlwolf-Z']} indicates non-mixed-unitaryness if the value is below zero, which is the case for all $p<1/3$. Both witnesses are normalized to $s=-1$ at $p=0$.
• Figure 3: Overview of the performance of the quantum memory witness (QM) and the Mendl-Wolf witness (MW) for convex combinations of different unital channels. The three channels $\mathcal{E}_{\text{LS}}$, $\mathcal{E}_{\text{AO}}$ and $\mathcal{E}_{\text{HMR}}$ are verifiably non-MU while the fully depolarizing channel $\mathcal{E}_{\text{D}}$ and the identity channel $\mathds{1}$ have known MU decompositions. It can be seen that for the convex combination of $\mathcal{E}_{\text{LS}}$ and $\mathcal{E}_{\text{D}}$ the QM witness (blue thin line) performs equally well as the Mendl-Wolf-witness (orange thick line), while it outperforms the latter one in every other of the investigated convex combinations. The solid gray lines indicate parameter ranges for which it is known that the channel is MU watrousMixingDoublyStochastic2009 while dashed gray lines indicate parameter ranges for which nothing can be concluded from the witnesses. The details and parameter ranges depicted in this figure can be found in App. \ref{['app:performance']} in Tab. \ref{['tab:overview_performance']}.
• Figure 4: Setting to realize irreversible unital two-qubit dynamics. The system-environment interaction is described by Eqs. \ref{['eq:twoqubit-unital']} and \ref{['eq:twoqubit-environment']}.
• Figure 5: Non-MU witness from Eq. \ref{['eq:objective']} applied to the dynamics in Fig. \ref{['fig:two-qubit-dynamics']}. The parameters $\kappa_2=0.4$, $\kappa_2=1.2$, $\Gamma_x=0.4$, $\Gamma_y=0$ and $\Gamma_z=1.0$ are chosen such that the dynamics has no mixed-unitary representation for almost all times HelStr2009.

Theorems & Definitions (5)

• Theorem 1
• proof
• Corollary 1
• proof
• proof