{"id":1009,"date":"2015-01-07T16:32:17","date_gmt":"2015-01-07T14:32:17","guid":{"rendered":"https:\/\/webs.uab.cat\/giq\/seminar\/recent-developments-asymptotic-quantum-hypothesis-testing\/"},"modified":"2015-01-07T16:32:17","modified_gmt":"2015-01-07T14:32:17","slug":"recent-developments-asymptotic-quantum-hypothesis-testing","status":"publish","type":"seminar","link":"https:\/\/webs.uab.cat\/giq\/seminar\/recent-developments-asymptotic-quantum-hypothesis-testing\/","title":{"rendered":"Recent developments in asymptotic quantum hypothesis testing."},"content":{"rendered":"
The issue of quantum state discrimination naturally arises in connection\r\nwith various application-oriented concepts from quantum information\r\nprocessing and quantum computation. The starting point is the general\r\nscenario where there is only incomplete  information about the state of\r\nthe quantum system at hand. Instead, there is a (finite) number of\r\nhypotheses about the possible quantum state. From an operational\r\nviewpoint, the task is to decide which hypothesis gives the best\r\ndescription of the true preparation of the quantum system if  a decision\r\n\n
The issue of quantum state discrimination naturally arises in connection\r\nwith various application-oriented concepts from quantum information\r\nprocessing and quantum computation. The starting point is the general\r\nscenario where there is only incomplete  information about the state of\r\nthe quantum system at hand. Instead, there is a (finite) number of\r\nhypotheses about the possible quantum state. From an operational\r\nviewpoint, the task is to decide which hypothesis gives the best\r\ndescription of the true preparation of the quantum system if  a decision\r\nrule has to be based on the outcomes of appropriate quantum measurements.\r\n\r\nIn the first part of the talk we will explain the associated mathematical\r\nproblem of quantum hypothesis testing where the aim is to minimize the\r\naveraged (Bayesian) error probability over the set of quantum tests\r\n(POVMs) for the given finite number of hypotheses. The focus should be on\r\nthe asymptotic setting where there is an arbitrary large number of copies\r\nof a finite-level quantum system available for being tested by means of\r\ncollective quantum measurements. In the special case of two simple quantum\r\nhypotheses, respectively described by a density operator, the optimal\r\nquantum tests are known to be given by the so-called Holevo-Helstrom\r\nprojectors and the corresponding maximal asymptotic error exponent has\r\nbeen identified with an almost closed expression usually referred to as\r\nquantum Chernoff distance. However, in the more general case of *<\/span>multiple*<\/span><\/strong>\r\n(more than two) quantum hypotheses similar general results are still\r\nmissing. Although, many partial results have been obtained in the recent\r\nyears. We will mention some of them in the talk, see for example [1,2,3] .\r\n\r\nIn the second part, we will present two new results in testing multiple\r\nsimple quantum hypotheses.  The first one provides asymptotically optimal\r\ntests in discriminating between almost pure states. The other one proves\r\nthat the ML-type quantum detectors introduced in [3]  asymptotically\r\nachieve the optimal error exponent equal to the multiple quantum Chernoff\r\nbound  subject to the so called mutually unbiasedness of the hypothetic\r\ndensity operators.\r\n\r\nIf time allows, in the third part of the talk we will introduce the\r\nextended problem of testing *<\/span>composite*<\/span><\/strong> quantum hypotheses, discuss the\r\nclose relation to multiple simple quantum hypothesis testing and provide\r\nasymptotically optimal solutions in some interesting special cases.\r\n\r\nReferences:\r\n\r\n1. Audenaert, K.M.R., Mosonyi, M. Upper bounds on the error probabilities\r\nand asymptotic error exponents in quantum multiple state discrimination,\r\nJ. Math. Phys. 55 , 102201 (2014)\r\n\r\n2. Nussbaum, M. and Szkola, A. Asymptotically optimal\r\ndiscrimination between multi- ple pure quantum states. In: Theory of\r\nQuantum Computation, Communication and Cryptography. 5th Conference, TQC\r\n2010, Leeds, UK. Revised Selected Papers. Lecture Notes in Computer\r\nScience, Vol 6519, van Dam, Wim; Kendon, Vivien M.; Severini, Simone\r\n(Eds.), pp. 1-8, Springer (2011)\r\n\r\n3. Nussbaum, M. and Szkola, A. An asymptotic error bound for\r\ntesting multiple quantum hypotheses, Ann. Statist. 39 No. 6 3211-3233\r\n(2011)\r\n<\/pre>\n","protected":false},"excerpt":{"rendered":"
The issue of quantum state discrimination naturally arises in connection with various application-oriented concepts from quantum information processing and quantum computation. The starting point is the general scenario where there is only incomplete information about the state of the quantum system at hand. Instead, there is a (finite) number of hypotheses about the possible quantum […]<\/p>\n","protected":false},"author":20,"featured_media":0,"template":"","class_list":["post-1009","seminar","type-seminar","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/webs.uab.cat\/giq\/wp-json\/wp\/v2\/seminar\/1009","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/webs.uab.cat\/giq\/wp-json\/wp\/v2\/seminar"}],"about":[{"href":"https:\/\/webs.uab.cat\/giq\/wp-json\/wp\/v2\/types\/seminar"}],"author":[{"embeddable":true,"href":"https:\/\/webs.uab.cat\/giq\/wp-json\/wp\/v2\/users\/20"}],"wp:attachment":[{"href":"https:\/\/webs.uab.cat\/giq\/wp-json\/wp\/v2\/media?parent=1009"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}