{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,7]],"date-time":"2025-11-07T13:20:13Z","timestamp":1762521613369,"version":"build-2065373602"},"reference-count":22,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2013,9,6]],"date-time":"2013-09-06T00:00:00Z","timestamp":1378425600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The correlation distance quantifies the statistical independence of two classical or quantum systems, via the distance from their joint state to the product of the marginal states. Tight lower bounds are given for the mutual information between pairs of two-valued classical variables and quantum qubits, in terms of the corresponding classical and quantum correlation distances. These bounds are stronger than the Pinsker inequality (and refinements thereof) for relative entropy. The classical lower bound may be used to quantify properties of statistical models that violate Bell inequalities. Partially entangled qubits can have lower mutual information than can any two-valued classical variables having the same correlation distance. The qubit correlation distance also provides a direct entanglement criterion, related to the spin covariance matrix. Connections of results with classically-correlated quantum states are briefly discussed.<\/jats:p>","DOI":"10.3390\/e15093698","type":"journal-article","created":{"date-parts":[[2013,9,6]],"date-time":"2013-09-06T12:09:54Z","timestamp":1378469394000},"page":"3698-3713","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Correlation Distance and Bounds for Mutual Information"],"prefix":"10.3390","volume":"15","author":[{"given":"Michael","family":"Hall","sequence":"first","affiliation":[{"name":"Centre for Quantum Computation and Communication Technology (Australian Research Council), Centre for Quantum Dynamics, Griffith University, Brisbane, QLD 4111, Australia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2013,9,6]]},"reference":[{"key":"ref_1","unstructured":"Cover, T.M., and Thomas, J.A. (2006). Elements of Information Theory, John Wiley & Sons. [2nd ed.]. Chapter 11."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1023\/A:1019653202562","article-title":"Approximate quantum error correction","volume":"1","author":"Schumacher","year":"2002","journal-title":"Quantum Inf. Process."},{"key":"ref_3","unstructured":"Hayashi, M. (2013). Large deviation analysis for classical and quantum security via approximate smoothing. Quantum Phys."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1109\/TIT.2012.2216952","article-title":"Strong secrecy and reliable Byzantine detection in the presence of an untrusted relay","volume":"59","author":"He","year":"2013","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1491","DOI":"10.1109\/TIT.2003.811927","article-title":"Refinements of Pinsker\u2019s inequality","volume":"49","author":"Fedotov","year":"2003","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"99","DOI":"10.2140\/pjm.1981.96.99","article-title":"Sufficiency, KMS conditions and relative entropy in von Neumann algebras","volume":"96","author":"Hiai","year":"1981","journal-title":"Pac. J. Math."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1895","DOI":"10.1007\/s11128-011-0347-6","article-title":"Fano type quantum inequalities in terms of q-entropies","volume":"11","author":"Rastegin","year":"2012","journal-title":"Quantum Inf. Process."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Brand\u00e3o, F.G.S.L., and Harrow, A.W. (2013). Quantum de Finetti theorems under local measurements with applications. Quantum Phys.","DOI":"10.1145\/2488608.2488718"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"3280","DOI":"10.1109\/TIT.2007.903122","article-title":"Estimating mutual information via Kolmogorov distance","volume":"53","author":"Zhang","year":"2007","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Aaronson, B., Franco, R.L., Compagno, G., and Adesso, G. (2013). Hierarchy and dynamics of trace distance correlations. Quantum Phys.","DOI":"10.1088\/1367-2630\/15\/9\/093022"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Paula, F.M., Montealegre, J.D., Saguia, A., de Oliveira, T.R., and Sarandy, M.S. (2013). Geometric classical and total correlations via trace distance. Quantum Phys.","DOI":"10.1209\/0295-5075\/103\/50008"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"022102","DOI":"10.1103\/PhysRevA.84.022102","article-title":"Relaxed Bell inequalities and Kochen-Specker theorems","volume":"84","author":"Hall","year":"2012","journal-title":"Phys. Rev. A"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"090502","DOI":"10.1103\/PhysRevLett.100.090502","article-title":"No-local-broadcasting theorem for multipartite quantum correlations","volume":"100","author":"Piani","year":"2008","journal-title":"Phys. Rev. Lett."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"187904","DOI":"10.1103\/PhysRevLett.91.187904","article-title":"Communication cost of simulating Bell correlations","volume":"91","author":"Toner","year":"2003","journal-title":"Phys. Rev. Lett."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"880","DOI":"10.1103\/PhysRevLett.23.880","article-title":"Proposed experiment to test local hidden-variable theories","volume":"23","author":"Clauser","year":"1969","journal-title":"Phys. Rev. Lett."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1007\/BF00417500","article-title":"Quantum generalizations of Bell\u2019s inequality","volume":"4","year":"1980","journal-title":"Lett. Math. Phys."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"855","DOI":"10.1103\/RevModPhys.55.855","article-title":"Pairs of two-level systems","volume":"55","author":"Fano","year":"1983","journal-title":"Rev. Mod. Phys."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Bhatia, R. (1997). Matrix Analysis, Springer-Verlag. Section IV.2.","DOI":"10.1007\/978-1-4612-0653-8"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"4277","DOI":"10.1103\/PhysRevA.40.4277","article-title":"Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model","volume":"40","author":"Werner","year":"1989","journal-title":"Phys. Rev. A"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"060301(R)","DOI":"10.1103\/PhysRevA.77.060301","article-title":"Entanglement detection beyond the computable cross-norm or realignment criterion","volume":"77","author":"Zhang","year":"2008","journal-title":"Phys. Rev. A"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"3824","DOI":"10.1103\/PhysRevA.54.3824","article-title":"Mixed-state entanglement and quantum error correction","volume":"54","author":"Bennett","year":"1996","journal-title":"Phys. Rev. A"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"032319","DOI":"10.1103\/PhysRevA.80.032319","article-title":"Correlations in local measurements on a quantum state, and complementarity as an explanation of nonclassicality","volume":"80","author":"Wu","year":"2009","journal-title":"Phys. Rev. A"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/15\/9\/3698\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T21:49:08Z","timestamp":1760219348000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/15\/9\/3698"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,9,6]]},"references-count":22,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2013,9]]}},"alternative-id":["e15093698"],"URL":"https:\/\/doi.org\/10.3390\/e15093698","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2013,9,6]]}}}