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Quantum Information Theory(QIT) is one of the most dynamic and exciting areas of research in science and technology. It overlaps many different fields of physics and mathematics such as Quantum Mechanics, Operator Theory, and Probability Theory. One of the most interesting branches of Probability Theory is Random Matrix Theory (RMT) and lately it has been proven that many ideas originating from RMT can be exploited for given reliable answers to the open question of QIT.
Our project focuses on a systematic exploration of theoretical questions in QIT using tools developed by the mathematicians and physicists working in Random Matrix Theory. The main goal of our three-year project is to systematically explore and provide answers for questions in QIT about random quantum states and random quantum channels. Such problems have attracted the attention lately in a very naturally connection to fundamental issues of QIT theory, such as entanglement theory and classical (or quantum) capacities for channels. Since explicit answers for non-trivial dimensions are hard (if not impossible) to give, physicists and computer scientists turned to typical states and channels, that is random states or channels. These examples proved to be very rich and they were the key ingredient to some very spectacular results, such as the non-additivity of the Holevo quantity for quantum channels or thresholds for the entanglement of typical mixed quantum states. Therefore, techniques coming from Random Matrix Theory can be considered natural tools to tackle the above problems. Another interesting example of this is the question of the additivity of the minimum output entropy (MOE) for a quantum channel. Quantum information theorists have shown that this question was equivalent to understanding the set of singular values of linear maps sampled from vector spaces of rectangular matrices. Hastings’ answer to this question, in the negative, does not shed light on how big the non -additivity can be or on what are the minimal dimensions in which this extraordinary phenomenon can occur. We believe that this question, natural from the perspective of RMT, can have a more precise answer and we are working towards a better understanding of the respective sets of singular values. Another problem of interest is to find the image of a finite set of states through random quantum channel. There are known attempts to solve this kind of problem using techniques coming from RMT. The image of classes of bipartite quantum states under a tensor product of random quantum channels have been derived lately. Interesting properties are found such as that the Bell state gives asymptotically, among some large classes of input states, the output with the least entropy. Such questions are relevant also in the framework of additive problems. Indeed, it does not prove that the Bell state gives the largest violation of additivity, but it stands as a solid mathematical evidence towards the fact that the physically intuitive choice of the Bell state is indeed close to being optimal.
Results obtained within RMTQIT project
TASK 1 : RANDOM QUANTUM CHANNELS: CAPACITIES AND THEIR IMAGE
Issue 1: convergence of the output set of random quantum channels
In the paper " On the convergence of output sets of quantum channels" ( arXiv:1311.7571), the authors Benoit Collins, Motohisa Fukuda, Ion Nechita show that the output set of random quantum channels admits a deterministic limit when the dimension of the input space grows. The work employs recent advances in the theory of random matrices to show that for fairly general models of randomness, the output set becomes non-random at the limit and it can be characterised by some norms arising in free probability. The authors apply then the main result to the case of random mixed quantum channels: these are channels which can be written as convex combinations of random unitary conjugations. These channels have been used and investigated extensively in quantum information theory and the authors provide, for the first time, a general framework to analyse their typical behaviour. Computing quantities of interest, such as minimum output Renyi entropies require however the computation of some norm which appears in free probability. The authors are able to do this only in the most simple case, that of the infinity (or operator) norm.
TASK 2 : TYPICALITY OF ENTANGLEMENT IN LARGE QUANTUM SYSTEMS
Issue 1: detect threshold points for Reduction Criterion (RC)
Using various techniques coming from random matrix theory, we approached the problem of finding threshold points for Reduction Criterion in different asymptotic regimes (balanced, unbalanced I, unbalanced II). Our explicit results are widely presented in our papers “On the reduction criterion for random quantum states'' (arXiv:1402.4292) and "Thresholds for reduction-related entanglement criteria in quantum information theory" (arXiv:1503.08008), co-authors Maria A. Jivulescu, Nicolae Lupa and Ion Nechita.
Issue 2: determine necessary and sufficient conditions for charactersing Absolutely Reduced (ARED) states.
The criterion of absolute reduction detects states, of a given spectrum, that satisfies RC. We achieved to give a characterisation of these states and to find explicit formulation of our conditions in
particular cases, such as pseudo-pure states. Moreover, lower and upper bounds for ARED, APPT states are found, by introducing new sets as GER or LS. Our results are published in the paper ``Positive reduction from spectra'' ( arXiv:1406.1277), co-authors Maria A. Jivulescu, Nicolae Lupa, Ion Nechita and David Reeb.
Issue 3: to detect threshold points for Absolute Reduction Criterion (ARC)
We aim at finding thresholds points for ARC, in different asymptotic regimes. These points are of interest from the point of view of the geometry of separable states. A complete picture of the status-of-art of the threshold points fir various entangled criteria is presented in our paper "Thresholds for reduction-related entanglement criteria in quantum information theory" (arXiv:1503.08008), co-authors Maria A. Jivulescu, Nicolae Lupa and Ion Nechita.
TASK 3 : RANDOM POSITIVE MAPS BETWEEN MATRIX ALGEBRAS
Issue 1: characterise channels with polytopic image
Ion Nechita, in collaboration with Motohisa Fukuda and Michael M. Wolf studied the particular class of quantum channels that have the following particularity: their image (the image of the set of quantum states) is a convex polytope. This class of channels contains the so-called ``classical-quantum'' channels, which display classical behaviour. Completely characterisation of channels with polytopic image is given, and it is shown that this class contains channels displaying some ``quantum'' behaviour, such as additivity violations. Also the class of channels which satisfy a stronger form of additivity, called image-additivity is characterised. These results are presented in "Quantum channels with polytopic images and image additivity'', (arXiv:1408.2340).
Issue 2: study k-positive maps
B. Collins, P. Hayden and Ion Nechita in "Random and free positive maps with applications to entanglement detection" (arXiv:1505.08042) studied random linear maps between matrix algebras. The authors introduced these maps starting from Choi isomorphism between linear applications and bipartite states. For any probability distribution supported on a compact interval, a sequence of random matrix is considered, invariant to unitary rotations, having as limit the given distribution. Using free probability theory, the authors succeed to find necessary and sufficient conditions such that the linear application to be positive or/and completely positive. The conditions are explicit, requiring additive convolutions. Also, the capacity of these maps is study in order to detect entanglement in bipartite quantum states.