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

Abstract

Quantum non-Markovianity in tripartite quantum states ${\rho }_{ABC}$ 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.

22+6 pages, Publised version

References in corpus (20)

• Steering, Entanglement, Nonlocality, and the EPR Paradox
• Quantum Non-Markovianity: Characterization, Quantification and Detection
• Measuring Quantum Coherence with Entanglement
• Distillation of secret key and entanglement from quantum states
• A complete family of separability criteria
• Structure of states which satisfy strong subadditivity of quantum entropy with equality
• Private Randomness Expansion With Untrusted Devices
• Operational Markov condition for quantum processes
• Fundamental bound on the reliability of quantum information transmission
• On the strong converses for the quantum channel capacity theorems
• One-Shot Operational Quantum Resource Theory
• Security of device-independent quantum key distribution protocols: a review
• Fundamental limits on quantum dynamics based on entropy change
• Modular commutator in gapped quantum many-body systems
• Conditional Mutual Information of Bipartite Unitaries and Scrambling
• Extendibility limits the performance of quantum processors
• Quantifying non-Markovianity: a quantum resource-theoretic approach
• Upper bounds on device-independent quantum key distribution rates in static and dynamic scenarios
• A paradigm for entanglement theory based on quantum communication
• Classical and quantum parts of conditional mutual information for open quantum systems

Cited by in corpus (1)

• Causal and Non-Causal Revivals of Information: A New Regime of Non-Markovianity in Quantum Stochastic Processes