### My 'Doctor of Sciences' dissertation

My research interests include

• quantum statistical decision theory: comparison of quantum experiments and channels, measurement and hypothesis testing on quantum channels and networks
• quantum divergences: constructions, inequalities and equality conditions, relations to sufficiency (reversibility) of channels
• quantum information geometry: finite dimensional and non-parametric
• mathematical foundations of quantum theory: effect algebras, synaptic algebras

Some more details and links to related papers are provided below.

## Quantum statistical decision theory

### Comparison of quantum channels and statistical experiments

The classical theory of comparison of statistical experiments was introduced in (D. Blackwell, Comparison of experiments, Proc. 2nd Berkeley Symp. on Math. Stat. and Probab., 1951, 93-102), where a preorder on statistical experiments was defined by comparison of average risks of decision rules. A deep result of this theory is the Blackwell-Sherman-Stein (BSS) theorem, which relates this preorder to randomization of experiments. An approximate version was given in (L. Le Cam, Sufficiency and approximate sufficiency, Ann. Math. 34, (1964), 1419-1455), introducing the notion of deficiency and a (pseudo-)distance of statistical experiments. The extension of the BSS theorem in this setting is the celebrated Le Cam's randomization criterion.

Inspired by the classical theory, we introduce a deficiency measure and a pseudo-distance for a pair of quantum channels with the same input state. This quantity expresses how precisely one channel can be approximated by post-processings of the other, in the diamond norm. We prove a quantum version of Le Cam's randomization criterion, showing that this distance can be determined by comparing optimal success probabilities of certain ensembles. We also study deficiency of quantum statistical experiments and obtain a quantum version of the randomization criterion.

### Quantum local asymptotic normality

In the joint work with Madalin Guta, we investigate a quantum version of the weak convergence of statistical experiments and the related local asymptotic normality. This means that a parametrized family of i.i.d. states of a quantum system can be locally approximated by a Gaussian shift, in the sense of weak convergence. For this, we introduce the notion of a canonical state of a quantum statistical experiment.

### Measurement and hypothesis testing on quantum channels

The set of qantum channels with a given input and output space has a natural convex structure, related to the possibility of choosing the channel at random. As it is the case in generalized probabilistic theories, a measurement on a convex set can be defined as an affine map, assigning to each element the corresponding probability distribution on the set of outcomes. The performance of measurements is then determined by the convex structure of the set. Since quantum channels can be represented as some (bipartite) states by the Choi isomorphism, measurements can be given in terms of certain operator valued measures, similar to POVMs.

Any measurement on channels can be implemented by applying the channels to an input state on the input space coupled with an ancilla and measuring the resulting states. In the joint work with Martin Plávala, we investigate optimality conditions in multiple hypothesis testing problems on quantum channels. We give conditions for existence of optimal tests with a maximally entangled input state. In the process, a new upper bound on the diamond norm is obtained.

As a generalization of measurements on the set of channels, quantum supermaps are defined as completely positive trace preserving maps on subspaces of a matrix algebra. These maps can be again represented by elements of some multipartite state space. In this way, we obtain a hierarchy of maps related to quantum combs, describing quantum networks (G. Chiribella et al., Theoretical framework for quantum networks, Phys. Rev. A, 80 (2009)). We investigate related norms on matrix algebras, their duals and their role in statistical decision theory for quantum supermaps.

The convex structure of sets of quantum devices is discused in the following papers:

## Quantum divergences and sufficiency of channels

The notion of a sufficient (or reversible) quantum channel with respect to a family of states was introduced in (D. Petz, Sufficiency of channels over von Neumann algebras, Quart. J. Math. Oxford, 39 (1988), 97-108), by analogy with cassical sufficient statistics. It means existence of a channel, called a recovery map, by which all states in the family can be recovered. This property is characterized in several equivalent ways, most importantly by equality in the data processing inequality (DPI) for some important entropic quantities, including the relative entropy, and by a certain structure of the channel and the involved states. This is used in finding conditions for equality in entropic inequalities, in particular for characterization of quantum Markov triples by equality in the strong subadditivity of entropy.

### Quantum sufficiency in the von Neumann algebra framework

In a joint work with Denes Petz, we investigated conditions for sufficiency of a normal two-positive map (coarse-graining) on a von Neumann algebra. The notion of a minimal sufficient subalgebra is introduced and a version of quantum factorization criterion is obtained in the type I case. Sufficiency is characterized by preservation of a version of quantum Fisher information and characterization of quantum Markov triples is extended to infinite dimensions.

### Quantum sufficiency in finite dimensional algebras

In finite dimensions, sufficiency conditions in terms of quasi-entropies and monotone Riemannian metrics (quantum Fisher information) are studied. Further, we investigate conditions related to hypothesis testing, such as $L_1$-distance, Chernoff bound and Hoeffding bound. In a joint paper with M. B. Ruskai, we study convexity and equality conditions for a class of trace functionals.

### Sandwiched Rényi relative entropies

Recently, a new, so-called sandwiched version of the quantum Rényi relative $\alpha$-entropy was introduced (M. Muller Lennert et al., On quantum Rényi entropies: a new generalization and some properties, J. Math. Phys., 54 (2013), 122203, M. M. Wilde et al., Strong converse for the classical capacity of entanglement-breaking and Hadamard channels via a sandwiched Rényi relative entropy, Commun. Math. Phys., 331 (2014), 593-622). This quantity is extended to the setting of arbitrary von Neumann algebras and it is shown that equality in DPI characterizes sufficiency of channels for all $\alpha>1$.

## Quantum information geometry

Classical information geometry studies the differentiable manifold structure of statistical models, the so-called information manifolds, see the monograph (S. Amari, H. Nagaoka, Method of Information Geometry, AMS Monograph, Oxford University Press (2000)). Any information manifold is endowed with a Riemannian metric, provided by the Fisher information, and a family of affine connections. These structures are essentially unique, respecting Markov morphisms between statistical models. Moreover, for dually flat manifolds, the dualistic properties of these structures yield natural divergence measures: the canonical divergences.

### Finite dimensional quantum case

It was first observed in (N. N. Cencov, E. A. Morozova, Markov invariant geometry on state manifolds, Itogi Nauki i Tekhniki, 36 (1990), 69-102) that the uniqueness no longer holds for quantum states: there is a large family of Riemannian metrics on the manifold of density matrices, which are monotone under quantum channels. Later, the monotone metrics were characterized by (D. Petz, Monotone metrics on matrix spaces, Lin. Alg. Appl., 244 (1996), 81-96). In the following works, affine connections on such manifolds are studied. We investigate their duality with respect to a given monotone metric and the possibility of introducing a dually flat affine structure. This requirement singles out a family of monotone metrics with a corresponding pair of dual connections.

### Nonparametric quantum information geometry

The nonparametric version of classical information manifolds was introduced by (G. Pistone, C.Sempi, An infinite-dimensional geometric structure on the space of all the probability measures equivalent to a given one, Ann. Statist., 23 (1995), 1543-1561). In this case, the manifold is modelled on a Banach space: the exponential Orlicz space. We introduce a quantum version of the eponential Orlicz space and show how this can be used to define a manifold structure on the set of faithful states of a von Neumann algebra. This construction is obtained from relative entropy and its relation to state perturbation. It is also shown that this structure behaves well under completely positive normal unital maps (channels), in particular, it is invariant if and only if the channel is sufficient (reversible) for the given set of states.

## Quantum observables

### Commutative POVMs and smearings of observables

In a joint work with Silvia Pulmannová and Elena Vinceková, we study smearings of observables on Hilbert spaces and also in the more general setting of effect algebras. A smearing of an observable is defined as a composition with a (weak) Markov kernel, in other words, a post-processing. It is proved that an observable is commutative if and only if it is a smearing of a sharp (or projection-valued) observable. Relations with previous characterization of commutative observables are given.

### Synaptic algebras

Synaptic algeras were introduced in (D. J. Foulis, Synaptic algebras, Mathematica Slovaca 60 (2010), 631-654) as a generalization of the set of bouded self-adjoint operators on a Hilbert space. This structure ties together the notions of a Jordan algebra, a spectral order-unit normed space, a convex effect algebra and an orthomodular lattice. In collaboration with D. Foulis and S. Pulmannová, we investigated some properties of operator algebras that remain true for synaptic algebras. Moreover, we studied states and observables on synaptic algebras and a version of the Loomis-Sikorski theorem for commutative synaptic algebras.