Quantum statistics: decision problems and entropic functionals on state spaces
Final Report Abstract
The general subject of the research project were statistical decision problems on state spaces of matrix algebras and large tensor products of them. The focus of the first work package was on quantum hypothesis testing extending the field of statistical hypothesis testing to the non-commutative ∗-algebraic framework which is used in quantum theory. It directly corresponds to the issue in physics which is the (minimal error) quantum state discrimination based on outcomes of generalized measurements. It is of relevance for novel computation and communication technologies relying on quantum systems. Apart from binary quantum hypothesis testing, for arbitrary finite sets of quantum states neither an explicit construction of quantum tests that minimize the averaged probability of erronuos state detection is known, nor sharp bounds on error rate exponents in the asymptotic setting, which aim to determine limitations as well as the power of optimal testing procedures, have been proven. In a work preceding the reported project we have introduced the multiple quantum Chernoff bound (qMCD) as a candidate for a sharp bound. Starting from that, within the first work package we have developed so called ML-type tests for multiple quantum hypotheses which coincide with the prominent maximum-likelihood decisions for sets of commuting density operators. Apart from the commuting case, these efficiently to construct quantum tests are not optimal. However, we succeeded to specify various sufficient conditions on sets of quantum states for asymptotic achievability of the qMCD bound by the ML-type tests. Further, we proposed quantum tests being a variant of the ML-type tests which, for arbitrary finite sets of states, achieve the bound at least up to a universal factor (1/3). The tightness of the qMCD bound remains an open question in the general case. Moreover, we have investigated testing problems between composite quantum hypotheses respectively respresented by a set of density operators. If one of two hypotheses is associated to a finite set of pure states we could verify the quantum Chernoff distance between the corresponding sets as the optimal asymptotic rate exponent of the averaged error probability. The focus of the second work package was on quantum statistical models being manifolds of quantum states which are known as Gibbs families in theoretical physics. Their elements are referred to as thermal states and satisfy a maximum-entropy principle under expected value constraints of one or more energy operators called Hamiltonians. The maximum-entropy inference with respect to a set of non-commutative Hamiltonians can have notable discontinuities (at the boundary of the set of expected values) which do not exist in the commutative case. For every Gibbs family we have proved a Pythagorean and a projection theorem which show for example that discontinuities in the maximum-entropy inference come from discrepancies between an information topology and the norm topology. A second explanation uses openness of the expected value functional which gives rise to continuity conditions in terms of convex geometry. While the starting point of this work package was the continuity analysis of the maximumentropy inference, we have also settled the longer standing problem about a geometric theory of many-body correlations introduced by Amari (2001) for probability vectors of full support. Ay et al. (2011) have extended the theory to probability vectors of arbitrary support and Zhou (2009) has established the quantum counterpart for density matrices of full support. The above mentioned Pythagorean and projection theorem have completed the quantum case of arbitrary support. It came as a surprise that Chen et al. have suggested in June 2014 that discontinuities of the maximum-entropy inference are signals of a quantum phase transition. This opens new perspectives for future research with application to condensed matter physics and topological quantum computation.
Publications
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(2011) An asymptotic error bound for testing quantum hypotheses, The Annals of Statistics 39(6) 3211-3233
Nussbaum, M., and Szkoła, A.
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(2012) Entropy distance: New quantum phenomena, Journal of Mathematical Physics 53(10) 102206
Weis, S., and Knauf, A.
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(2014) Continuity of the maximum-entropy inference, Communications in Mathematical Physics 330(3) 1263–1292
Weis, S.
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(2014) Information topologies on non-commutative state spaces, Journal of Convex Analysis 21(2) 339–399
Weis, S.
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Maximizing the divergence from a hierarchical model of quantum states
Weis, S., Knauf, A., Ay, N., and Zhao, M.-J.
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Continuity of the maximum-entropy inference: convex geometry and numerical ranges approach
Rodman, L., Spitkovsky, I. M., Szkoła, A., and Weis, S.
