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

Rajeev Gangwar; Tanmoy Pandit; Kaumudibikash Goswami; Siddhartha Das; Manabendra Nath Bera

doi:10.22331/q-2025-02-26-1646

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

Rajeev Gangwar¹, Tanmoy Pandit², Kaumudibikash Goswami³, Siddhartha Das⁴, and Manabendra Nath Bera¹

¹Department of Physical Sciences, Indian Institute of Science Education and Research (IISER), Mohali, Punjab 140306, India
²Fritz Haber Research Center for Molecular Dynamics, Hebrew University of Jerusalem, Jerusalem 9190401, Israel
³QICI Quantum Information and Computation Initiative, Department of Computer Science, The University of Hong Kong, Pokfulam Road, Hong Kong
⁴Center for Security, Theory and Algorithmic Research (CSTAR), Centre for Quantum Science and Technology (CQST), International Institute of Information Technology, Hyderabad, Gachibowli, Telangana 500032, India

Published:	2025-02-26, volume 9, page 1646
Editor:	Cyril Branciard
Eprint:	arXiv:2311.18323v3
Doi:	https://doi.org/10.22331/q-2025-02-26-1646
Citation:	Quantum 9, 1646 (2025).

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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.

Featured image: The resource-free operations that cannot increase or create genuine quantum non-Markovianity in a tripartite quantum system ($ABC$) where $B$ is the conditional system. Here, the LOQC (LOCC) means local operations and quantum (classical) communication, while the arrows indicate the directions of communication.

Popular summary

In classical information theory, probability distribution $p_{XYZ}$ is a Markov chain $X-Y-Z$ if the mutual information between random variables $X$ and $Z$ conditioned on $Y$ vanishes. Quantum Markov chain states are defined analogously as tripartite states $\rho_{ABC}$ such that its quantum conditional mutual information $I(A;C|B)_{\rho}$ vanishes. There exists a set of quantum Markov chain states whose probabilistic mixture is not a quantum Markov chain state. We identify such non-Markovianity as that of classical origin.

We introduce a new information-theoretic measure to characterize non-Markovianity in a tripartite quantum state of genuine quantum origin. This measure reveals a novel tripartite quantum correlation and may be interpreted as entanglement between two systems when conditioned on the third (memory) system. We formulate a convex resource-theoretic framework to characterize genuine quantum non-Markovianity in states as a quantum resource and provide operational meaning to this resource in various information processing tasks.

► BibTeX data

@article{Gangwar2025squashedquantumnon,
  doi = {10.22331/q-2025-02-26-1646},
  url = {https://doi.org/10.22331/q-2025-02-26-1646},
  title = {Squashed quantum non-{M}arkovianity: a measure of genuine quantum non-{M}arkovianity in states},
  author = {Gangwar, Rajeev and Pandit, Tanmoy and Goswami, Kaumudibikash and Das, Siddhartha and Bera, Manabendra Nath},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1646},
  month = feb,
  year = {2025}
}

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[1] Francesco Buscemi, Rajeev Gangwar, Kaumudibikash Goswami, Himanshu Badhani, Tanmoy Pandit, Brij Mohan, Siddhartha Das, and Manabendra Nath Bera, "Causal and Noncausal Revivals of Information: A New Regime of Non-Markovianity in Quantum Stochastic Processes", PRX Quantum 6 2, 020316 (2025).

[2] Siddhartha Das, Kaumudibikash Goswami, and Vivek Pandey, "Conditional entropy and information of quantum processes", arXiv:2410.01740, (2024).

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