I work in the broad area of quantum information theory and I am interested in problems which are mathematical in nature. Entanglement is one of the fundamental concepts in quantum physics and quantum information. As one moves from pure to mixed states and to Hilbert spaces of dimensions larger than 2, a plethora of unsolved problems begin to emerge. The characterization of entanglement is important on the one hand because of the fundamental role it plays in distinguishing the quantum from classical, and on the other hand because of its practical use in quantum computation. My interest lies in the geometry of quantum states and structure of quantum entanglement, with a particular emphasis on PPT entangled states. In this context, we have discovered new classes of PPT entangled states, new ways of extending entanglement witnesses and interesting connections between different methods of characterizing entanglement of PPT states. Research Contribution.
Research Contribution: In my work, I look for discovering new classes of entangled states. To detect entanglement, I use the approach of positive maps which are not completely positive (CP). This connects the above problem to a long standing problem in C* algebra:- what is the structure theorem of all positive maps between two given C* algebras, A and B. For CP maps, we have the Steinspring dilation theorem, but nothing such is known for positive maps in general. It can be shown that for the mappings \mathcal{B}(\mathbb{C}^m)\longrightarrow\mathcal{B}(\mathbb{C}^n) where (m, n) is one of the following - (2, 2), (2, 3), (3, 2), all positive maps are decomposable i.e. can be written as a sum of a CP and transpose of a CP map. Thus for finite dimensional quantum systems 2 ⊗ 2, 2 ⊗ 3 and 3⊗2, we have a necessary and sufficient condition to detect entanglement. A quantum state ρ is separable if and only if it remains positive under partial transpose (i.e. using transpose in one of the system, keeping the other fixed, or mathematically, using the map 1 ⊗ T where 1 is the identity map and T is transpose). But for no other dimensions, positivity under partial transpose (PPT) is a sufficient condition. Indeed in all other dimensions, there exist PPT entangled states. Hence the maps which can detect such states have to be in-decomposable. The first such example of such maps was found by Choi. Till now there are only a few examples of such maps. Most of them are modification of Choi’s example. To understand entanglement, it is also important to detect the structure theorem of such positive not CP and in-decomposable maps. The last point is in particular difficult to detect.
Since the set of such maps forms a closed convex set (bounded also if restricted to some sub classes, like unital), it is enough to find out the extreme points of such sets. Indeed, these extreme points (here maps), are more powerful in detecting entangled states than interior points. Choi’s map is one example of such extreme objects. In general, it is difficult to determine the set of all such objects. There is no method to check, whether given a positive map is extremal or not, which would have settled the question of extremal property of many such available examples. Indeed the bigger goal of my research is to get a Krein-Milman theory for positive maps. An analogous theory in the case of CP maps is a fertile field of research for both functional analysis and quantum mechanics.
In our work, we have developed methods for generating such extremal points/ maps. We also have shown that they have interesting properties in terms of detecting entangled states. Given a positive in-decomposable map, we have invented a numerical method to generate a PPT entangled state which can be detected by that map. A modified version of the method, can generate the PPT state undetectable by a finite set of maps of our choice. In an upcoming work, we have further generalised the above method to generate these extreme maps. Given a positive map on B(Cn ), we combined it with n-super-positive maps and thus generate the positive maps. We have shown the necessary and sufficient conditions to make these combination positive but not CP, in-decomposable, and extremal. We have noticed that, such extremal maps can have same or different powers in terms of detecting entanglement, and detected the criteria for having same power. We have used the modified Choi map and shown that these maps can be used for detecting PPT entangled states arising from unextendible product basis (UBP).
As mentioned earlier, given a positive map, there is no method to check whether it is extremal or not. We are in the process of developing a method which, in a limited way can detect extremal points of a given class of positive maps.
Future Research Plans: The concept of operator space and completely bounded maps has become an important object in recent years. It is to be noted that all the above examples of maps, except transpose, are completely bounded. I want to explore the above problems by using operator space techniques (and also in an abstract C* -algebra settings). It is obvious that such maps can be written as a difference of two CP maps. But the difference of two arbitrary CP maps need not be CP. So we need to look for tight conditions to make such objects positive. Following works of Størmer, we are trying to determine the exposed properties of such objects and are also trying to extend the above techniques in a more general set up. With my training in Mathematics at the Masters level and a PhD in Quantum Information Theory, I now want to embark upon looking at problems in this area which have a mathematical flavour and I am open to learning about and working on newer issues within this general framework.