# Research problems of the project

#### Project » Research problems of the project

The state-of-the-art described above provides a motivation to perform research on problems tabularized as work packages below.

**WP1: Control of Generation of Resources for Non-Classical Communication and Metrology**

Entangled states distributed between partner provide them with strong correlations between outcomes of local measurements which cannot be reconstructed by the means of local actions and classical communication. Examples of such tasks include teleportation of a quantum state [Bennett1993], reduction of communication complexity, dense coding and high sensitivity quantum metrology; a review can be found in [Pan2008]. These applications require mobile quantum information carriers. A natural choice is to use photons. Today, the most commonly used source of entangled photons is the process known as spontaneous parametric down-conversion (PDC), and techniques related to entanglement swapping, which allow to observe multi-photon interference effects (GHZ states, multi photon singlets, etc) The strength of these correlations depends on physical parameters of the setup, such as the spectral and the spatial properties of the beam, its power, or a shape and dimensions of the crystal. Our aim is to analyze in depth these factors for various configurations, and various types of primary sources.

*WP1.1: Analysis of noise structure and its dependence on external parameters*. We want to investigate how ma- nipulating of the parameters affects the noise observed in experiments. Given an experimental setup we want to investigate what state is seen exactly by detectors collecting photons from the source. We want to recognize the range of parameters, for which the state can be used as a resource in a specific communication task.

*WP1.2: Enhancement of non-classical properties by manipulation of noise*. We want to study ways of improving the non-classicality of products of parametric down-conversion by active control of the characteristic of the pump beam, the various types of PDC sources, and the down-converted photons. Most of attention will be paid to post- processing a down-converted state. Techniques such as individual filtering or re-imposing a required symmetry will be investigated.

**WP2: Detailed Structure of Sets of Entangled States.**

Today, protocols utilizing quantum resources go beyond simple two-qubit ones. This provokes questions about general relations between subsystems of a composite quantum system. Currently such relations are known mainly for collections of qubits. Another fact is that by local operations and classical communication not all entangled states are transformable into some specific state. Understanding of this structure will allow us to effectively classify resources for quantum communication with respect to their specific use. It will also be possible to design quantum information processing protocols, in which only a subset of partners will have to collaborate for success.

*WP2.1: Correlations in higher-dimensional systems*. The aim of this task is to formulate the complementarity relations for states of many high-dimensional quantum systems. For qubits, such relations are well known. Their easiest form is given in terms of violation of Bell inequalities. Similar conditions for higher-dimensional systems would allow to predict limitations of next generations of quantum protocols. The task includes finding new Bell inequalities for such system, studies on formulation of conditions for their violation, and their combining in order to retrieve mutual dependencies. The task may also cover criteria for the possibility of transformations between various classes of entangled states.

*WP2.2: Non-classicality of globally uncorrelated states*. Recently, the community's attention was drawn toward states, which do not possess correlations between odd numbers of parties. Such states appear as reduced states in generation of Dicke or multiphoton singlet states. We want to investigate the conditions these state must satisfy to be genuinely multipartite entangled, or to possess a specific kind of correlations (e.g. they are transformable by LOCC to GHZ states). Additionally, we would like to prove that the reduced states of multiphoton Dicke or singlet states have applications in quantum information processing.

*WP2.3: Entanglement for metrology*. Possible applications of entanglement in metrology are still not fully investigated. A standard problem in metrology is the phase estimation in an interferometer. It was proved, that if we can estimate the phase more precisely with a quantum state than classically, this state is entangled. It was never established, however, what kind of entanglement is responsible for this advantage. There are large groups of entangled states, which do not provide the advantage. We also want to study the phase estimation precision with multi-mode interferometers (and higher-dimensional states), and non-linear phase-shifters.

**WP3: New Tools for Entanglement Detection**

Modern theory of quantum information processing describes phenomena and protocols involving many particles. Also recent experimental realizations allow to observe quantum correlations in for many quantum systems. This progress, however, comes at the price of precision. Using many entangled pairs simultaneously causes a drastic drop of parameters such as visibility or fidelity. Additionally, emissions of a larger number of photons occur and this impairs the statistical quality of data. Note that a tomography of a state of a collection of qubits requires an exponentially large amount of data. Therefore there is a need for entanglement criteria involving less measurement effort.

*WP3.1: Optimal entanglement detection.* We want to construct criteria, which for a given set of measurements could distinguish as many entangled states from separable states as possible. This will limit the requirement of optimizing the measurements for a successful entanglement detection.

*WP3.2: Economic entanglement detection*. Symmetries of a physical problem allow much simpler analysis. If a state possesses even an approximate symmetry, it greatly reduces the number of required measurements to determine it with a high precision. Some of invariances have already been studied in this context, for example, with respect to the permutation symmetry. It is also possible to simplify the analysis of almost pure states (with one strongly dominating eigenvalue). We want to investigate entanglement criteria for states with new symmetries. One of examples from the laboratories is a situation, in which qubits are split into two groups with a permutation symmetry. The trade-off between optimal and economic criteria shall be investigated.

**WP4: Quantum information and thermodynamics of micro and nano-systems.**

Our main goal is to get fundamental limitations for the systems such as working bodies which are microscopic. We shall consider two main directions:

1. (bottom up) Contribute to clarifying the picture of thermodynamics for microscopic systems, by considering some generic models of system-bath coupling.

2. (top down) Develop thermodynamics as a resource theory, by use of the methods of quantum information; using the latter approach, obtain fundamental limits for processing micro-systems in presence of a heat bath.

The two approaches are of an opposite nature, the first one is more down to Earth, while the second has a more axiomatic nature. We hope that they can be smoothly reconciled. This will be also our aim. Moreover, we shall develop resource theories involving superselection rules.

*WP4.1:* We want to consider situation, in which both the working body and the system that collects work are quantum. As mentioned, even the question of defining drawing work in micro-regime is far from clear. We aim to develop further this subject, in order to obtain clear picture of thermodynamics of microscopic systems. To this end we shall try to devise dynamical setups involving heat bath, and micro-systems playing role of working bodies and the work storage.

*WP4.2:* As proposed in [Horodecki2011] the application of formalism of [Janzing2000] to the situation, in which a heat bath is treated as a free resource opens a new path of research, leading to understanding what transformations are possible in presence of heat bath. This includes probing fundamental limits for drawing work, and manipulating systems in presence of heat bath(s) in various configurations. While we have preliminary results for manipulating with many identical copies of a state (call it 'macro' regime), we plan to obtain develop rules of transformations in 'meso' and 'micro' regime, that would generalize the rules obtained in [Horodecki2003]. We then hope to apply the rules to analyze Carnot cycle with microscopic working-body, and other more complicated scenarios. We also aim to reconcile a definition of work proposed in [Horodecki2011] related to having a system in a pure excited state, with the macroscopic definition of work, based on energy stored in a classical field.

**WP5: Mathematical aspects of quantum correlations and quantum formalism in general**

The programme centers around important themes in mathematical physics and operator algebras. The main goal is the application of powerful functional analysis techniques to study quantum maps on properly defined spaces of ob- servables. More specifically the important challenge is to give a precise mathematical treatment (and classification) of the interesting quantum maps that arise from the quantum physics. The concepts of quantum Orlicz-spaces, operator spaces, positivity, entanglement and finally non-commutative entropy all play a part in this programme. The effort in this programme will be focused around the following basic themes:

*WP5.1: Analysis of positive maps on properly chosen spaces of (regular) observables.* For several decades, mathematical physics has been developing into a coherent methodology capable of application to a wide variety of theoretical models of physical systems. Building a rigorous mathematical theory has proved to be a hard challenge, especially for dynamical systems. However, the need for such a theory has been so great that much high-grade effort has been devoted to it. Many crucial ideas are now in place and we are beginning to study the hardest problems in the field. A nice example of such old open problems is the classification of positive maps on quantum structures [Woronowicz1976]. We have for some time now studied in detail (non-)decomposable positive maps on C* (Jordan) algebras, with a special emphasis on low dimensional cases [Majewski2007] as well as on the space associated with regular quantum statistical systems [Labuschange2001]. We hope to continue with this programme by building on the expertise already gained, in order to obtain greater clarity regarding the structure of these maps. Our main tools will be operator theory and quantum probability as they provide the mathematical framework and language to describe quantum dynamical systems.

*WP5.2: Quantum entropy and quantum correlations*. It seems that the concept of quantum correlations is one of the most intriguing quantum phenomena leading to various quantum effects. We would like to examine fundamental principles governing creation, manipulation and observation of correlations. Having an experience with an analysis of regular quantum statistical systems we wish to study the new approach to an analysis of regular quantum states. In particular, working within the setting of noncommutative structures, we want to employ the mentioned approach for getting the "correct" quantization of dynamical entropy. As entropy" measures the disorder" of a system, there is a hope that this concept will be extremely useful for studying ergodic properties of quantum dynamical systems. The participants of the programme have been active in the subarea dealing with mathematical aspects of dynami- cal systems, description of quantum correlations and the structure of PPT states [Majewski2004-1, Majewski2004-2, Majewski2009]. The goal of the programme is to try and bundle this expertise.

*WP5.3: Quantum analysis*. The important point to note here is the well known fact that classical (quantum) Koopmanism leads to basic ingredients

of classical (quantum respectively) ergodic theory [Antoniou2002]. The ongoing challenge of quantum analysis is to as far as possible quantize classical theory by extending it to the context of noncommutative versions of the above mentioned spaces. In this way one ultimately arrives at a quantized version of classical analysis where the non-commutative integration theory plays the crucial role. The significance of such a quantized analysis is that it provides a rigorous mathematical framework for studying quantum phenomena, and for doing much of the modeling discussed elsewhere in this description. Of late there has been an emerging realization that quantum Orlicz spaces potentially have a very important role to play in various aspects of Quantum Theory [Labuschange2011].

**WP6: Direct collaboration with experimental laboratories**

Although the previously listed Work Packages, especially 1-3, would in many case lead to new experimental ideas, this one is a list of problems/experiments that we would like to work on hand-in-hand with our experimental colleagues from the very beginning. The main objective is to extend knowledge about entanglement and to give new experimental schemes which use entangled photons, both in applications, as well as to generate new non-classical multi-photon effects. The aims are:

- new states of entangled photons,
- new methods of quantum communication,
- experimental tests of such methods,
- measures of non-classicality.

The aims are divided into two groups. The first group is the research concerning production and analysis of states of light (various types of multi-photon entanglement, which involve more then just two qubits, entanglement of higher dimensional systems). The theoretical aims will concentrate on classification of the new states and their characterization. The experimental realizations will employ the best work horse" of quantum optics: parametric down conversion. However, we could move to other sources, like quantum dots or search for more optimal down conversion sources (see the problem of entangled frequencies of the twin photons), as well a entanglement swapping methods allowing to entangle remote atoms in traps (with Weinfurter). The second group of aims is the application of the new states in quantum communication. The new traits of the states would enable new methods of commu- nication. One can expect better effciency and capacity of channels, as well as realization of systems which involve more than two parties. It would be also interesting to pinpoint fundamental limitations, which are imposed by the real structure of experimental realizations of the quantum information processes. Some specific problems:

- new multi-photon entangled states
- new communication complexity problems [see e.g., Trojek2005] which allow for quantum protocols
- what are the implications of the close relation between communication complexity problems and Bell's theorem, as well as the links with secrete sharing protocols; as well as implications a new version of temporal inequalities (2010),
- design of new experiments which would demonstrate new ways to reduce communication complexity
- application of the new Bell inequalities in multipartite quantum cryptography schemes (secret sharing),
- application of the graph" states in demonstrations of fundamental quantum computational tasks,
- simplification of linear optics quantum gates,
- foundational issues behind quantum information processing (e.g. Information Causality, etc.).
- quantum information tasks with continuous variables
- novel uses of known optical processes, like e.g. in weak measurements.

All these problems are open. In some cases, like e. g. quantum cryptography with many partners, the basic principles are known, however, the full range of possibilities given by multi-particle states is still unknown. Some of the posed questions grow out of earlier problems researched by the author of the project. Thus, there is a good basis (experience) for the new research. The aim is not a verification of known ideas, but finding yet unknown applications and properties of quantum states, and new methods in quantum information processing. Note, that a realization of these proposals is highly dependent of foreign laboratories. In some cases the ideas could be realized in Polish laboratories like e.g. KL FAMO (Torun), or even at University of Gdask, where an acousto-optical interference experiment involving entangled PDC photons is being carried out. We would try to encourage experimental groups to make at the early stage several optical experiments. These could include:

- Communication complexity and secret sharing experimental schemes involving many partners and only pairs of entangled photons. Recently, with the group of Weinfurter we have designed, analyzed and performed realization of (quantum) communication complexity and secret sharing schemes in which partners pass form one to another a single qubit [Trojek2005]. However, if one replaces the qubit by a pair of entangled qubits, one immediately gains in the security of the scheme (as the individual photons do not have a fixed polarization).
- As big part of our research in the coming years will concentrate around the problem of quantum repeaters, we would strongly encourage this line of experimental research. One should attempt to create at least one full high-fidelity stage station for a quantum repeater. A lot of interesting spin-off experiments could result from this effort.
- An initial stage of the above effort would be a continuation of the work toward perfecting the interference experiment(s) involving photons originating from independent sources. In this case an effort should be made to get photon indistinguishability not only via suitable filtering, but to tailor the PDC sources in such a way that already the initial idler-signal frequency correlations are weak, or almost non-existent.
- Phase conjugating mirrors, just like parametric down-conversion sources, if just pumped by two counter- propagating fields emit noise" which is in the form of spontaneously emitted pairs of photons. It would be interesting to investigate this effect in experiment, as an alternative to PDC.

The existing method of computer analysis of the non-classicality of quantum correlations given in [Kaszlikowski2001] has not as yet exhausted all its potential applications, and could be used to analyze new quantum states, as well as experimental data (by the way, the very problem of a direct analysis of raw experimental data by our program is a very interesting one. As it was observed by Gisin (private communication) one has to find specific methods to eliminate the uctuations in local frequencies of counts, as these are misinterpreted by the program as non-local influences). We have recently made a progress in our computer code employed for studying these matters, so a new series of results can be expected.

Finally the ramifications of the recent discovery of a new form of Temporal Bell inequalities should be intensively studied, as this may shed new light on the root of the quantum speedup in many elementary information processing tasks, which do not necessarily involve entanglement. As a closing note of the presentation of the tasks, we would like to stress that all that was said above are our current plans, based on current knowledge. In case of emergence of important new ideas related to the general topic of this project (quantum technologies), we would at least change the accents of our scientific effort, or try to follow new directions with full force.