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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 (<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>B<\/mml:mi><\/mml:math>) to Alice (<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>A<\/mml:mi><\/mml:math>) or Charlie (<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>C<\/mml:mi><\/mml:math>) 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.<\/jats:p>","DOI":"10.22331\/q-2025-02-26-1646","type":"journal-article","created":{"date-parts":[[2025,2,26]],"date-time":"2025-02-26T11:11:07Z","timestamp":1740568267000},"page":"1646","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":1,"title":["Squashed quantum non-Markovianity: a measure of genuine quantum non-Markovianity in states"],"prefix":"10.22331","volume":"9","author":[{"given":"Rajeev","family":"Gangwar","sequence":"first","affiliation":[{"name":"Department of Physical Sciences, Indian Institute of Science Education and Research (IISER), Mohali, Punjab 140306, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tanmoy","family":"Pandit","sequence":"additional","affiliation":[{"name":"Fritz Haber Research Center for Molecular Dynamics, Hebrew University of Jerusalem, Jerusalem 9190401, Israel"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Kaumudibikash","family":"Goswami","sequence":"additional","affiliation":[{"name":"QICI Quantum Information and Computation Initiative, Department of Computer Science, The University of Hong Kong, Pokfulam Road, Hong Kong"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Siddhartha","family":"Das","sequence":"additional","affiliation":[{"name":"Center for Security, Theory and Algorithmic Research (CSTAR), Centre for Quantum Science and Technology (CQST), International Institute of Information Technology, Hyderabad, Gachibowli, Telangana 500032, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Manabendra Nath","family":"Bera","sequence":"additional","affiliation":[{"name":"Department of Physical Sciences, Indian Institute of Science Education and Research (IISER), Mohali, Punjab 140306, India"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"9598","published-online":{"date-parts":[[2025,2,26]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"A. 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