Recent developments in asymptotic quantum hypothesis testing.

Recent developments in asymptotic quantum hypothesis testing.

Seminar author:Arleta Szkola

Event date and time:01/29/2015 02:30:pm

Event location:IFAE seminar room

Event contact:

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 state. From an operational
viewpoint, the task is to decide which hypothesis gives the best
description of the true preparation of the quantum system if  a decision

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 state. From an operational
viewpoint, the task is to decide which hypothesis gives the best
description of the true preparation of the quantum system if  a decision
rule has to be based on the outcomes of appropriate quantum measurements.

In the first part of the talk we will explain the associated mathematical
problem of quantum hypothesis testing where the aim is to minimize the
averaged (Bayesian) error probability over the set of quantum tests
(POVMs) for the given finite number of hypotheses. The focus should be on
the asymptotic setting where there is an arbitrary large number of copies
of a finite-level quantum system available for being tested by means of
collective quantum measurements. In the special case of two simple quantum
hypotheses, respectively described by a density operator, the optimal
quantum tests are known to be given by the so-called Holevo-Helstrom
projectors and the corresponding maximal asymptotic error exponent has
been identified with an almost closed expression usually referred to as
quantum Chernoff distance. However, in the more general case of *multiple*
(more than two) quantum hypotheses similar general results are still
missing. Although, many partial results have been obtained in the recent
years. We will mention some of them in the talk, see for example [1,2,3] .

In the second part, we will present two new results in testing multiple
simple quantum hypotheses.  The first one provides asymptotically optimal
tests in discriminating between almost pure states. The other one proves
that the ML-type quantum detectors introduced in [3]  asymptotically
achieve the optimal error exponent equal to the multiple quantum Chernoff
bound  subject to the so called mutually unbiasedness of the hypothetic
density operators.

If time allows, in the third part of the talk we will introduce the
extended problem of testing *composite* quantum hypotheses, discuss the
close relation to multiple simple quantum hypothesis testing and provide
asymptotically optimal solutions in some interesting special cases.

References:

1. Audenaert, K.M.R., Mosonyi, M. Upper bounds on the error probabilities
and asymptotic error exponents in quantum multiple state discrimination,
J. Math. Phys. 55 , 102201 (2014)

2. Nussbaum, M. and Szkola, A. Asymptotically optimal
discrimination between multi- ple pure quantum states. In: Theory of
Quantum Computation, Communication and Cryptography. 5th Conference, TQC
2010, Leeds, UK. Revised Selected Papers. Lecture Notes in Computer
Science, Vol 6519, van Dam, Wim; Kendon, Vivien M.; Severini, Simone
(Eds.), pp. 1-8, Springer (2011)

3. Nussbaum, M. and Szkola, A. An asymptotic error bound for
testing multiple quantum hypotheses, Ann. Statist. 39 No. 6 3211-3233
(2011)