Squashed quantum non-Markovianity: a measure of genuine quantum non-Markovianity in states

TL;DR

The paper introduces squashed quantum non-Markovianity (sQNM) to quantify genuine quantum non-Markovian correlations in tripartite states by minimizing quantum conditional mutual information over all extensions, thereby removing classical contributions. It develops a convex resource theory with free states (vanishing sQNM) and free operations (not increasing sQNM) and proves key properties such as convexity, monogamy, asymptotic continuity, and faithfulness, relating sQNM to squashed entanglement and state extendibility. It then establishes fundamental limits on state transformations and quantum communication costs in terms of changes in sQNM, and provides operational meanings for sQNM through two tasks: a variant of conditional one-time pad (rate = 2 $N_{ m sq}$) and state deconstruction (cost = $I(A;C|B)$, or $2N_{ m sq}$ with extensions). Collectively, these results position sQNM as a robust quantum resource that captures quantum-origin non-Markovianity and connects it to practical information-processing tasks and the structure of quantum correlations.

Abstract

Quantum non-Markovianity in tripartite quantum states $ρ_{ABC}$ represents a correlation between systems $A$ and $C$ when conditioned on the system $B$ and is known to have both classical and quantum contributions. However, a systematic characterization of the latter is missing. To address this, we propose a faithful measure for non-Markovianity of genuine quantum origin called squashed quantum non-Markovianity (sQNM). It is based on the quantum conditional mutual information and is defined by the left-over non-Markovianity after squashing out all non-quantum contributions. It is lower bounded by the squashed entanglement between non-conditioning systems in the reduced state and is delimited by the extendibility of either of the non-conditioning systems. We show that the sQNM is monogamous, asymptotically continuous, convex, additive on tensor-product states, and generally super-additive. We characterize genuine quantum non-Markovianity as a resource via a convex resource theory after identifying free states with vanishing sQNM and free operations that do not increase sQNM in states. We use our resource-theoretic framework to bound the rate of state transformations under free operations and to study state transformation under non-free operations; in particular, we find the quantum communication cost from Bob ($B$) to Alice ($A$) or Charlie ($C$) is lower bounded by the change in sQNM in the states. The sQNM finds operational meaning; in particular, the optimal rate of private communication in a variant of conditional one-time pad protocol is twice the sQNM. Also, the minimum deconstruction cost for a variant of quantum deconstruction protocol is given twice the sQNM of the state.

Figures (2)

• Figure 1: Set of states with vanishing sQNM. Here, $\mathcal{F}^{X}_S$ refers to the convex set of free states with the optimal extension for $N_{\mathrm{sq}}{=}0$ achieved through a classical extension, see Eq. \ref{['eq:ClassExt']}. Outside of $\mathcal{F}^{X}_S$, there exist free states for which $N_{\mathrm{sq}}{=}0$ is achieved via extensions beyond classical registers, see Observation \ref{['obs:QuantExt']}. In general, we represent the convex set of free states with the most general extension by $\mathcal{F}^{E}_S$, with the set $\mathcal{F}^{X}_S$ contained within it, i.e., $\mathcal{F}^{X}_S\subsetneq \mathcal{F}^{E}_S$ holds.
• Figure 2: The figure depicts the free operations under which the sQNM is monotonically non increasing. Here, we assume Alice, Bob, and Charlie possess the systems $\mathbf{A}$, $\mathbf{B}$, and $\mathbf{C}$, respectively, and Bob is used as the conditioning system. The arrows represent the direction of communication while implementing the operations. The free operations are local operation and classical communication between Alice and Charlie $\mathbb{LOCC}_{\mathbf{A} \leftrightarrow \mathbf{C}}$, involving both-way classical communications; and local operation and quantum communication $\mathbb{LOQC}_{\mathbf{A} \to \mathbf{B}}$ ($\mathbb{LOQC}_{\mathbf{C} \to \mathbf{B}}$) involving one-way quantum communication from Alice (Charlie) to Bob, while no classical communication is allowed from Bob to Alice (Charlie). See Lemma \ref{['prop:FreeOps']} for more details.

Theorems & Definitions (17)

• Definition 1: Squashed quantum non-Markovianity
• Lemma 1
• Lemma 2
• theorem 1
• Lemma 3: Free operations
• Definition 2
• Definition 3
• Remark 1
• theorem 2: Converse bound
• theorem 3: Achievable bound